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## G = (C2×C12).55D4order 192 = 26·3

### 29th non-split extension by C2×C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12).55D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — (C2×C12).55D4
 Lower central C3 — C22×C6 — (C2×C12).55D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).55D4
G = < a,b,c,d | a2=b12=c4=1, d2=ab6, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 328 in 134 conjugacy classes, 59 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×C12, C2×C12, C22×C6, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×C12, C23.83C23, C6.C42, C6.C42, C2×C4⋊Dic3, C6×C4⋊C4, (C2×C12).55D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×Dic6, C4○D12, D42S3, Q83S3, C2×C3⋊D4, C23.83C23, C4.Dic6, C4⋊C4⋊S3, C12.48D4, C23.23D6, C12.23D4, (C2×C12).55D4

Smallest permutation representation of (C2×C12).55D4
Regular action on 192 points
Generators in S192
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 191)(8 192)(9 181)(10 182)(11 183)(12 184)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 97)(24 98)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 109)(35 110)(36 111)(37 168)(38 157)(39 158)(40 159)(41 160)(42 161)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 145)(60 146)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 177)(68 178)(69 179)(70 180)(71 169)(72 170)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 85)(82 86)(83 87)(84 88)(121 144)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)(129 140)(130 141)(131 142)(132 143)
(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 20 50 43)(2 105 51 161)(3 18 52 41)(4 103 53 159)(5 16 54 39)(6 101 55 157)(7 14 56 37)(8 99 57 167)(9 24 58 47)(10 97 59 165)(11 22 60 45)(12 107 49 163)(13 155 48 192)(15 153 38 190)(17 151 40 188)(19 149 42 186)(21 147 44 184)(23 145 46 182)(25 72 134 83)(26 169 135 86)(27 70 136 81)(28 179 137 96)(29 68 138 79)(30 177 139 94)(31 66 140 77)(32 175 141 92)(33 64 142 75)(34 173 143 90)(35 62 144 73)(36 171 133 88)(61 122 84 111)(63 132 74 109)(65 130 76 119)(67 128 78 117)(69 126 80 115)(71 124 82 113)(85 114 180 125)(87 112 170 123)(89 110 172 121)(91 120 174 131)(93 118 176 129)(95 116 178 127)(98 156 166 181)(100 154 168 191)(102 152 158 189)(104 150 160 187)(106 148 162 185)(108 146 164 183)
(1 90 191 80)(2 89 192 79)(3 88 181 78)(4 87 182 77)(5 86 183 76)(6 85 184 75)(7 96 185 74)(8 95 186 73)(9 94 187 84)(10 93 188 83)(11 92 189 82)(12 91 190 81)(13 144 105 127)(14 143 106 126)(15 142 107 125)(16 141 108 124)(17 140 97 123)(18 139 98 122)(19 138 99 121)(20 137 100 132)(21 136 101 131)(22 135 102 130)(23 134 103 129)(24 133 104 128)(25 159 118 46)(26 158 119 45)(27 157 120 44)(28 168 109 43)(29 167 110 42)(30 166 111 41)(31 165 112 40)(32 164 113 39)(33 163 114 38)(34 162 115 37)(35 161 116 48)(36 160 117 47)(49 174 153 70)(50 173 154 69)(51 172 155 68)(52 171 156 67)(53 170 145 66)(54 169 146 65)(55 180 147 64)(56 179 148 63)(57 178 149 62)(58 177 150 61)(59 176 151 72)(60 175 152 71)

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

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

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

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 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - - + image C1 C2 C2 C2 S3 D4 Q8 D6 C4○D4 Dic6 C3⋊D4 C4○D12 D4⋊2S3 Q8⋊3S3 kernel (C2×C12).55D4 C6.C42 C2×C4⋊Dic3 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 5 1 1 1 2 2 3 10 4 4 4 2 2

Matrix representation of (C2×C12).55D4 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
,
 12 3 0 0 0 0 8 1 0 0 0 0 0 0 4 0 0 0 0 0 5 10 0 0 0 0 0 0 6 0 0 0 0 0 0 11
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 11 8 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 5 12 0 0 0 0 0 0 11 8 0 0 0 0 11 2 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,4,5,0,0,0,0,0,10,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,11,1,0,0,0,0,8,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,5,0,0,0,0,0,12,0,0,0,0,0,0,11,11,0,0,0,0,8,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

(C2×C12).55D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{55}D_4
% in TeX

G:=Group("(C2xC12).55D4");
// GroupNames label

G:=SmallGroup(192,545);
// by ID

G=gap.SmallGroup(192,545);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,344,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a*b^6,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=b^6*c^-1>;
// generators/relations

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