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## G = (C2×C12)⋊3C8order 192 = 26·3

### 1st semidirect product of C2×C12 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — (C2×C12)⋊3C8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C22×C3⋊C8 — (C2×C12)⋊3C8
 Lower central C3 — C6 — (C2×C12)⋊3C8
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for (C2×C12)⋊3C8
G = < a,b,c | a2=b12=c8=1, ab=ba, ac=ca, cbc-1=ab5 >

Subgroups: 200 in 118 conjugacy classes, 75 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C2×C42, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C22×C12, C22×C12, C22.7C42, C22×C3⋊C8, C2×C4×C12, (C2×C12)⋊3C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C3⋊C8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C3⋊C8, C4.Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C22.7C42, C4×C3⋊C8, C42.S3, C12⋊C8, C12.55D4, C6.C42, (C2×C12)⋊3C8

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

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

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

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

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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C4 C4 C8 S3 D4 Q8 Dic3 D6 M4(2) C3⋊C8 Dic6 C4×S3 D12 C3⋊D4 C4.Dic3 kernel (C2×C12)⋊3C8 C22×C3⋊C8 C2×C4×C12 C2×C3⋊C8 C22×C12 C2×C12 C2×C42 C2×C12 C2×C12 C22×C4 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 8 4 16 1 3 1 2 1 4 8 2 4 2 4 8

Matrix representation of (C2×C12)⋊3C8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 27 0 0 0 0 66 66 0 0 7 59
,
 63 0 0 0 0 72 0 0 0 0 43 43 0 0 13 30
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,27,0,0,0,0,66,7,0,0,66,59],[63,0,0,0,0,72,0,0,0,0,43,13,0,0,43,30] >;

(C2×C12)⋊3C8 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_3C_8
% in TeX

G:=Group("(C2xC12):3C8");
// GroupNames label

G:=SmallGroup(192,83);
// by ID

G=gap.SmallGroup(192,83);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,184,6278]);
// 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=a*b^5>;
// generators/relations

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