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## G = (C2×C4).44D12order 192 = 26·3

### 37th non-split extension by C2×C4 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4).44D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — (C2×C4).44D12
 Lower central C3 — C22×C6 — (C2×C4).44D12
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4).44D12
G = < a,b,c,d | a2=b12=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 360 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C4⋊C4, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×C12, C23.81C23, C6.C42, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C6×C4⋊C4, (C2×C4).44D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×D12, C4○D12, D42S3, S3×Q8, C2×C3⋊D4, C23.81C23, Dic3.Q8, C4.D12, C127D4, C23.23D6, Dic3⋊Q8, (C2×C4).44D12

Smallest permutation representation of (C2×C4).44D12
Regular action on 192 points
Generators in S192
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 133)(24 134)(25 158)(26 159)(27 160)(28 161)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 157)(37 117)(38 118)(39 119)(40 120)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 121)(59 122)(60 123)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 97)(71 98)(72 99)(85 180)(86 169)(87 170)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 164 66 48)(2 30 67 115)(3 162 68 46)(4 28 69 113)(5 160 70 44)(6 26 71 111)(7 158 72 42)(8 36 61 109)(9 168 62 40)(10 34 63 119)(11 166 64 38)(12 32 65 117)(13 181 176 122)(14 150 177 58)(15 191 178 132)(16 148 179 56)(17 189 180 130)(18 146 169 54)(19 187 170 128)(20 156 171 52)(21 185 172 126)(22 154 173 50)(23 183 174 124)(24 152 175 60)(25 99 110 77)(27 97 112 75)(29 107 114 73)(31 105 116 83)(33 103 118 81)(35 101 120 79)(37 82 165 104)(39 80 167 102)(41 78 157 100)(43 76 159 98)(45 74 161 108)(47 84 163 106)(49 133 153 91)(51 143 155 89)(53 141 145 87)(55 139 147 85)(57 137 149 95)(59 135 151 93)(86 129 140 188)(88 127 142 186)(90 125 144 184)(92 123 134 182)(94 121 136 192)(96 131 138 190)
(1 17 83 139)(2 16 84 138)(3 15 73 137)(4 14 74 136)(5 13 75 135)(6 24 76 134)(7 23 77 133)(8 22 78 144)(9 21 79 143)(10 20 80 142)(11 19 81 141)(12 18 82 140)(25 49 158 124)(26 60 159 123)(27 59 160 122)(28 58 161 121)(29 57 162 132)(30 56 163 131)(31 55 164 130)(32 54 165 129)(33 53 166 128)(34 52 167 127)(35 51 168 126)(36 50 157 125)(37 188 117 146)(38 187 118 145)(39 186 119 156)(40 185 120 155)(41 184 109 154)(42 183 110 153)(43 182 111 152)(44 181 112 151)(45 192 113 150)(46 191 114 149)(47 190 115 148)(48 189 116 147)(61 173 100 90)(62 172 101 89)(63 171 102 88)(64 170 103 87)(65 169 104 86)(66 180 105 85)(67 179 106 96)(68 178 107 95)(69 177 108 94)(70 176 97 93)(71 175 98 92)(72 174 99 91)

G:=sub<Sym(192)| (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,133)(24,134)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,157)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,121)(59,122)(60,123)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,97)(71,98)(72,99)(85,180)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,164,66,48)(2,30,67,115)(3,162,68,46)(4,28,69,113)(5,160,70,44)(6,26,71,111)(7,158,72,42)(8,36,61,109)(9,168,62,40)(10,34,63,119)(11,166,64,38)(12,32,65,117)(13,181,176,122)(14,150,177,58)(15,191,178,132)(16,148,179,56)(17,189,180,130)(18,146,169,54)(19,187,170,128)(20,156,171,52)(21,185,172,126)(22,154,173,50)(23,183,174,124)(24,152,175,60)(25,99,110,77)(27,97,112,75)(29,107,114,73)(31,105,116,83)(33,103,118,81)(35,101,120,79)(37,82,165,104)(39,80,167,102)(41,78,157,100)(43,76,159,98)(45,74,161,108)(47,84,163,106)(49,133,153,91)(51,143,155,89)(53,141,145,87)(55,139,147,85)(57,137,149,95)(59,135,151,93)(86,129,140,188)(88,127,142,186)(90,125,144,184)(92,123,134,182)(94,121,136,192)(96,131,138,190), (1,17,83,139)(2,16,84,138)(3,15,73,137)(4,14,74,136)(5,13,75,135)(6,24,76,134)(7,23,77,133)(8,22,78,144)(9,21,79,143)(10,20,80,142)(11,19,81,141)(12,18,82,140)(25,49,158,124)(26,60,159,123)(27,59,160,122)(28,58,161,121)(29,57,162,132)(30,56,163,131)(31,55,164,130)(32,54,165,129)(33,53,166,128)(34,52,167,127)(35,51,168,126)(36,50,157,125)(37,188,117,146)(38,187,118,145)(39,186,119,156)(40,185,120,155)(41,184,109,154)(42,183,110,153)(43,182,111,152)(44,181,112,151)(45,192,113,150)(46,191,114,149)(47,190,115,148)(48,189,116,147)(61,173,100,90)(62,172,101,89)(63,171,102,88)(64,170,103,87)(65,169,104,86)(66,180,105,85)(67,179,106,96)(68,178,107,95)(69,177,108,94)(70,176,97,93)(71,175,98,92)(72,174,99,91)>;

G:=Group( (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,133)(24,134)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,157)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,121)(59,122)(60,123)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,97)(71,98)(72,99)(85,180)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,164,66,48)(2,30,67,115)(3,162,68,46)(4,28,69,113)(5,160,70,44)(6,26,71,111)(7,158,72,42)(8,36,61,109)(9,168,62,40)(10,34,63,119)(11,166,64,38)(12,32,65,117)(13,181,176,122)(14,150,177,58)(15,191,178,132)(16,148,179,56)(17,189,180,130)(18,146,169,54)(19,187,170,128)(20,156,171,52)(21,185,172,126)(22,154,173,50)(23,183,174,124)(24,152,175,60)(25,99,110,77)(27,97,112,75)(29,107,114,73)(31,105,116,83)(33,103,118,81)(35,101,120,79)(37,82,165,104)(39,80,167,102)(41,78,157,100)(43,76,159,98)(45,74,161,108)(47,84,163,106)(49,133,153,91)(51,143,155,89)(53,141,145,87)(55,139,147,85)(57,137,149,95)(59,135,151,93)(86,129,140,188)(88,127,142,186)(90,125,144,184)(92,123,134,182)(94,121,136,192)(96,131,138,190), (1,17,83,139)(2,16,84,138)(3,15,73,137)(4,14,74,136)(5,13,75,135)(6,24,76,134)(7,23,77,133)(8,22,78,144)(9,21,79,143)(10,20,80,142)(11,19,81,141)(12,18,82,140)(25,49,158,124)(26,60,159,123)(27,59,160,122)(28,58,161,121)(29,57,162,132)(30,56,163,131)(31,55,164,130)(32,54,165,129)(33,53,166,128)(34,52,167,127)(35,51,168,126)(36,50,157,125)(37,188,117,146)(38,187,118,145)(39,186,119,156)(40,185,120,155)(41,184,109,154)(42,183,110,153)(43,182,111,152)(44,181,112,151)(45,192,113,150)(46,191,114,149)(47,190,115,148)(48,189,116,147)(61,173,100,90)(62,172,101,89)(63,171,102,88)(64,170,103,87)(65,169,104,86)(66,180,105,85)(67,179,106,96)(68,178,107,95)(69,177,108,94)(70,176,97,93)(71,175,98,92)(72,174,99,91) );

G=PermutationGroup([[(1,83),(2,84),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,141),(20,142),(21,143),(22,144),(23,133),(24,134),(25,158),(26,159),(27,160),(28,161),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,168),(36,157),(37,117),(38,118),(39,119),(40,120),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,121),(59,122),(60,123),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,97),(71,98),(72,99),(85,180),(86,169),(87,170),(88,171),(89,172),(90,173),(91,174),(92,175),(93,176),(94,177),(95,178),(96,179),(145,187),(146,188),(147,189),(148,190),(149,191),(150,192),(151,181),(152,182),(153,183),(154,184),(155,185),(156,186)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,164,66,48),(2,30,67,115),(3,162,68,46),(4,28,69,113),(5,160,70,44),(6,26,71,111),(7,158,72,42),(8,36,61,109),(9,168,62,40),(10,34,63,119),(11,166,64,38),(12,32,65,117),(13,181,176,122),(14,150,177,58),(15,191,178,132),(16,148,179,56),(17,189,180,130),(18,146,169,54),(19,187,170,128),(20,156,171,52),(21,185,172,126),(22,154,173,50),(23,183,174,124),(24,152,175,60),(25,99,110,77),(27,97,112,75),(29,107,114,73),(31,105,116,83),(33,103,118,81),(35,101,120,79),(37,82,165,104),(39,80,167,102),(41,78,157,100),(43,76,159,98),(45,74,161,108),(47,84,163,106),(49,133,153,91),(51,143,155,89),(53,141,145,87),(55,139,147,85),(57,137,149,95),(59,135,151,93),(86,129,140,188),(88,127,142,186),(90,125,144,184),(92,123,134,182),(94,121,136,192),(96,131,138,190)], [(1,17,83,139),(2,16,84,138),(3,15,73,137),(4,14,74,136),(5,13,75,135),(6,24,76,134),(7,23,77,133),(8,22,78,144),(9,21,79,143),(10,20,80,142),(11,19,81,141),(12,18,82,140),(25,49,158,124),(26,60,159,123),(27,59,160,122),(28,58,161,121),(29,57,162,132),(30,56,163,131),(31,55,164,130),(32,54,165,129),(33,53,166,128),(34,52,167,127),(35,51,168,126),(36,50,157,125),(37,188,117,146),(38,187,118,145),(39,186,119,156),(40,185,120,155),(41,184,109,154),(42,183,110,153),(43,182,111,152),(44,181,112,151),(45,192,113,150),(46,191,114,149),(47,190,115,148),(48,189,116,147),(61,173,100,90),(62,172,101,89),(63,171,102,88),(64,170,103,87),(65,169,104,86),(66,180,105,85),(67,179,106,96),(68,178,107,95),(69,177,108,94),(70,176,97,93),(71,175,98,92),(72,174,99,91)]])

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 S3 Q8 D4 D6 C4○D4 D12 C3⋊D4 C4○D12 D4⋊2S3 S3×Q8 kernel (C2×C4).44D12 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 3 2 1 1 1 4 4 3 6 4 4 4 2 2

Matrix representation of (C2×C4).44D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 8 5
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 7 0 0 0 0 10 10

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,7,10] >;

(C2×C4).44D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{44}D_{12}
% in TeX

G:=Group("(C2xC4).44D12");
// GroupNames label

G:=SmallGroup(192,540);
// by ID

G=gap.SmallGroup(192,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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