# Coset decomposition of a group by a subgroup

crystod-group --coset --point-group m-3m --subgroup 4/mmm

* Groups *
 G = m-3m (order 48)
 H = 4/mmm (order 16)

* Coset decomposition G = sum_i g_i H *
 index [G:H] = 3
 coset 1 (representative: E):
   { E, C4#1, C4#2, C4^2#1, C4^2#2, C4^2#3, C2#1, C2#4, i, S4#1, S4#2, sgh#1, sgh#2, sgh#3, sgd#4, sgd#1 }
 coset 2 (representative: C3#1):
   { C3#1, C2#3, C4#5, C3#7, C3#3, C3#5, C4#6, C2#6, S6#1, sgd#3, S4#5, S6#7, S6#3, S6#5, sgd#6, S4#6 }
 coset 3 (representative: C3#2):
   { C3#2, C4#4, C2#2, C3#8, C3#4, C3#6, C4#3, C2#5, S6#2, S4#4, sgd#2, S6#8, S6#4, S6#6, sgd#5, S4#3 }



# Space-group mode: decompose G by the little co-group G_k
# (one coset per arm of the star of k)
crystod-group --coset --space-group Pm-3m --kpoint 0.5 0.5 0

* Space group *
 Pm-3m (221)

* k point (primitive) *
 [0.5, 0.5, 0.0]

* Little co-group G_k *
 order |G_k| = 16 (|G| = 48)
 { 1, 2_100, 2_010, 2_001, 4^+_001, 4^-_001, 2_110, 2_-110, -1, m_100, m_010, m_001, -4^+_001, -4^-_001, m_110, m_-110 }

* Coset decomposition G = sum_i G_k g_i (one coset per arm of the star) *
 index [G:G_k] = |star of k| = 3
 coset 1 (representative: 1, k arm = [+0.5, +0.5, +0]):
   { 1, 2_100, 2_010, 2_001, 4^+_001, 4^-_001, 2_110, 2_-110, -1, m_100, m_010, m_001, -4^+_001, -4^-_001, m_110, m_-110 }
 coset 2 (representative: 3^+_111, k arm = [+0.5, +0, +0.5]):
   { 3^+_111, 3^+_1-1-1, 3^+_-11-1, 3^+_-1-11, 4^+_100, 4^-_100, 2_011, 2_01-1, -3^+_111, -3^+_1-1-1, -3^+_-11-1, -3^+_-1-11, -4^+_100, -4^-_100, m_011, m_01-1 }
 coset 3 (representative: 3^-_111, k arm = [+0, +0.5, +0.5]):
   { 3^-_111, 3^-_1-1-1, 3^-_-11-1, 3^-_-1-11, 4^+_010, 4^-_010, 2_101, 2_-101, -3^-_111, -3^-_1-1-1, -3^-_-11-1, -3^-_-1-11, -4^+_010, -4^-_010, m_101, m_-101 }
