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## G = C2.(C4×D12)  order 192 = 26·3

### 1st central stem extension by C2 of C4×D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2.(C4×D12)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — C2.(C4×D12)
 Lower central C3 — C2×C6 — C2.(C4×D12)
 Upper central C1 — C23 — C2.C42

Generators and relations for C2.(C4×D12)
G = < a,b,c,d | a2=b4=c12=1, d2=a, cbc-1=dbd-1=ab=ba, ac=ca, ad=da, dcd-1=c-1 >

Subgroups: 352 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.63C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2.(C4×D12)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, S3×C2×C4, C2×D12, C4○D12, D42S3, S3×Q8, C23.63C23, C4×D12, C23.16D6, C23.8D6, C23.21D6, Dic6⋊C4, Dic3.Q8, C4.D12, C2.(C4×D12)

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D4 D6 C4○D4 C4×S3 D12 C4○D12 D4⋊2S3 S3×Q8 kernel C2.(C4×D12) C6.C42 C3×C2.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 C4⋊Dic3 C2.C42 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 3 1 1 1 1 8 1 2 2 3 8 4 4 4 3 1

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

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 11 0 0 0 0 0 5 0 0 0 0 0 0 1 11 0 0 0 0 1 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 11 1 0 0 0 0 10 2 0 0 0 0 0 0 8 0 0 0 0 0 8 5 0 0 0 0 0 0 10 0 0 0 0 0 0 4
,
 3 5 0 0 0 0 11 10 0 0 0 0 0 0 8 10 0 0 0 0 0 5 0 0 0 0 0 0 0 4 0 0 0 0 10 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[11,10,0,0,0,0,1,2,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[3,11,0,0,0,0,5,10,0,0,0,0,0,0,8,0,0,0,0,0,10,5,0,0,0,0,0,0,0,10,0,0,0,0,4,0] >;

C2.(C4×D12) in GAP, Magma, Sage, TeX

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

G:=Group("C2.(C4xD12)");
// GroupNames label

G:=SmallGroup(192,212);
// by ID

G=gap.SmallGroup(192,212);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,387,58,6278]);
// Polycyclic

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

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