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## G = C4⋊Dic12order 192 = 26·3

### The semidirect product of C4 and Dic12 acting via Dic12/Dic6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4⋊Dic12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — C4⋊Dic12
 Lower central C3 — C6 — C2×C12 — C4⋊Dic12
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4⋊Dic12
G = < a,b,c | a4=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 296 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, Dic12, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C42Q16, C2.Dic12, C3×C4⋊C8, C4×Dic6, C122Q8, C2×Dic12, C4⋊Dic12
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×Q16, C8.C22, Dic12, C2×D12, S3×D4, Q83S3, C42Q16, C12⋊D4, C2×Dic12, C8.D6, C4⋊Dic12

Smallest permutation representation of C4⋊Dic12
Regular action on 192 points
Generators in S192
(1 75 192 33)(2 34 169 76)(3 77 170 35)(4 36 171 78)(5 79 172 37)(6 38 173 80)(7 81 174 39)(8 40 175 82)(9 83 176 41)(10 42 177 84)(11 85 178 43)(12 44 179 86)(13 87 180 45)(14 46 181 88)(15 89 182 47)(16 48 183 90)(17 91 184 25)(18 26 185 92)(19 93 186 27)(20 28 187 94)(21 95 188 29)(22 30 189 96)(23 73 190 31)(24 32 191 74)(49 139 111 167)(50 168 112 140)(51 141 113 145)(52 146 114 142)(53 143 115 147)(54 148 116 144)(55 121 117 149)(56 150 118 122)(57 123 119 151)(58 152 120 124)(59 125 97 153)(60 154 98 126)(61 127 99 155)(62 156 100 128)(63 129 101 157)(64 158 102 130)(65 131 103 159)(66 160 104 132)(67 133 105 161)(68 162 106 134)(69 135 107 163)(70 164 108 136)(71 137 109 165)(72 166 110 138)
(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)
(1 145 13 157)(2 168 14 156)(3 167 15 155)(4 166 16 154)(5 165 17 153)(6 164 18 152)(7 163 19 151)(8 162 20 150)(9 161 21 149)(10 160 22 148)(11 159 23 147)(12 158 24 146)(25 97 37 109)(26 120 38 108)(27 119 39 107)(28 118 40 106)(29 117 41 105)(30 116 42 104)(31 115 43 103)(32 114 44 102)(33 113 45 101)(34 112 46 100)(35 111 47 99)(36 110 48 98)(49 89 61 77)(50 88 62 76)(51 87 63 75)(52 86 64 74)(53 85 65 73)(54 84 66 96)(55 83 67 95)(56 82 68 94)(57 81 69 93)(58 80 70 92)(59 79 71 91)(60 78 72 90)(121 176 133 188)(122 175 134 187)(123 174 135 186)(124 173 136 185)(125 172 137 184)(126 171 138 183)(127 170 139 182)(128 169 140 181)(129 192 141 180)(130 191 142 179)(131 190 143 178)(132 189 144 177)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 4 12 12 12 12 24 24 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + - + - - + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 Q16 C4○D4 D12 Dic12 C8.C22 S3×D4 Q8⋊3S3 C8.D6 kernel C4⋊Dic12 C2.Dic12 C3×C4⋊C8 C4×Dic6 C12⋊2Q8 C2×Dic12 C4⋊C8 Dic6 C2×C12 C42 C2×C8 C12 C12 C2×C4 C4 C6 C4 C4 C2 # reps 1 2 1 1 1 2 1 2 2 1 2 4 2 4 8 1 1 1 2

Matrix representation of C4⋊Dic12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 32 39 0 0 0 0 28 41
,
 64 0 0 0 0 0 2 8 0 0 0 0 0 0 0 25 0 0 0 0 35 32 0 0 0 0 0 0 0 2 0 0 0 0 36 0
,
 7 23 0 0 0 0 36 66 0 0 0 0 0 0 61 55 0 0 0 0 4 12 0 0 0 0 0 0 61 31 0 0 0 0 26 12

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,32,28,0,0,0,0,39,41],[64,2,0,0,0,0,0,8,0,0,0,0,0,0,0,35,0,0,0,0,25,32,0,0,0,0,0,0,0,36,0,0,0,0,2,0],[7,36,0,0,0,0,23,66,0,0,0,0,0,0,61,4,0,0,0,0,55,12,0,0,0,0,0,0,61,26,0,0,0,0,31,12] >;

C4⋊Dic12 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_{12}
% in TeX

G:=Group("C4:Dic12");
// GroupNames label

G:=SmallGroup(192,408);
// by ID

G=gap.SmallGroup(192,408);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,226,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^4=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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