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## G = C2×C12⋊C8order 192 = 26·3

### Direct product of C2 and C12⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C12⋊C8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C12⋊C8
 Lower central C3 — C6 — C2×C12⋊C8
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C12⋊C8
G = < a,b,c | a2=b12=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 216 in 138 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C4⋊C8, C2×C42, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C22×C12, C2×C4⋊C8, C12⋊C8, C22×C3⋊C8, C2×C4×C12, C2×C12⋊C8
Quotients:

Smallest permutation representation of C2×C12⋊C8
Regular action on 192 points
Generators in S192
(1 153)(2 154)(3 155)(4 156)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 118)(14 119)(15 120)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 165)(26 166)(27 167)(28 168)(29 157)(30 158)(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 85)(57 86)(58 87)(59 88)(60 89)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 177)(68 178)(69 179)(70 180)(71 169)(72 170)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 97)(83 98)(84 99)(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 55 143 27 190 83 16 176)(2 54 144 26 191 82 17 175)(3 53 133 25 192 81 18 174)(4 52 134 36 181 80 19 173)(5 51 135 35 182 79 20 172)(6 50 136 34 183 78 21 171)(7 49 137 33 184 77 22 170)(8 60 138 32 185 76 23 169)(9 59 139 31 186 75 24 180)(10 58 140 30 187 74 13 179)(11 57 141 29 188 73 14 178)(12 56 142 28 189 84 15 177)(37 107 112 63 156 93 123 164)(38 106 113 62 145 92 124 163)(39 105 114 61 146 91 125 162)(40 104 115 72 147 90 126 161)(41 103 116 71 148 89 127 160)(42 102 117 70 149 88 128 159)(43 101 118 69 150 87 129 158)(44 100 119 68 151 86 130 157)(45 99 120 67 152 85 131 168)(46 98 109 66 153 96 132 167)(47 97 110 65 154 95 121 166)(48 108 111 64 155 94 122 165)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 8A ··· 8P 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - - + - + - + image C1 C2 C2 C2 C4 C4 C8 S3 D4 Q8 Dic3 D6 Dic3 D6 M4(2) C3⋊C8 Dic6 D12 C4.Dic3 kernel C2×C12⋊C8 C12⋊C8 C22×C3⋊C8 C2×C4×C12 C4×C12 C22×C12 C2×C12 C2×C42 C2×C12 C2×C12 C42 C42 C22×C4 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 2 1 4 4 16 1 2 2 2 2 2 1 4 8 4 4 8

Matrix representation of C2×C12⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 1 0 0 0 72 1 0 0 0 0 0 7 66 0 0 0 7 14
,
 63 0 0 0 0 0 44 31 0 0 0 2 29 0 0 0 0 0 36 31 0 0 0 68 37

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,1,1,0,0,0,0,0,7,7,0,0,0,66,14],[63,0,0,0,0,0,44,2,0,0,0,31,29,0,0,0,0,0,36,68,0,0,0,31,37] >;

C2×C12⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes C_8
% in TeX

G:=Group("C2xC12:C8");
// GroupNames label

G:=SmallGroup(192,482);
// by ID

G=gap.SmallGroup(192,482);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,100,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^12=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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