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## G = Dic12⋊C4order 192 = 26·3

### 3rd semidirect product of Dic12 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic12⋊C4
G = < a,b,c | a24=c4=1, b2=a12, bab-1=a-1, cac-1=a13, bc=cb >

Subgroups: 264 in 108 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C24, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, Dic12, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, Q16⋊C4, C2.Dic12, C8⋊Dic3, C3×C8⋊C4, C4×Dic6, C2×Dic12, Dic12⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C8.C22, S3×C2×C4, C2×D12, C4○D12, Q16⋊C4, C4×D12, C8.D6, Dic12⋊C4

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G ··· 4N 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 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 12 ··· 12 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

42 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 D4 D6 D6 C4○D4 C4×S3 D12 C4○D12 C8.C22 C8.D6 kernel Dic12⋊C4 C2.Dic12 C8⋊Dic3 C3×C8⋊C4 C4×Dic6 C2×Dic12 Dic12 C8⋊C4 C2×C12 C42 C2×C8 C12 C8 C2×C4 C4 C6 C2 # reps 1 2 1 1 2 1 8 1 2 1 2 2 4 4 4 2 4

Matrix representation of Dic12⋊C4 in GL6(𝔽73)

 65 0 0 0 0 0 34 9 0 0 0 0 0 0 56 19 25 70 0 0 27 2 38 22 0 0 48 3 17 54 0 0 35 51 46 71
,
 3 38 0 0 0 0 19 70 0 0 0 0 0 0 58 21 2 3 0 0 32 15 40 71 0 0 2 3 58 21 0 0 40 71 32 15
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(73))| [65,34,0,0,0,0,0,9,0,0,0,0,0,0,56,27,48,35,0,0,19,2,3,51,0,0,25,38,17,46,0,0,70,22,54,71],[3,19,0,0,0,0,38,70,0,0,0,0,0,0,58,32,2,40,0,0,21,15,3,71,0,0,2,40,58,32,0,0,3,71,21,15],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

Dic12⋊C4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,275);
// by ID

G=gap.SmallGroup(192,275);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,387,58,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=c^4=1,b^2=a^12,b*a*b^-1=a^-1,c*a*c^-1=a^13,b*c=c*b>;
// generators/relations

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