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## G = Dic6⋊C8order 192 = 26·3

### 3rd semidirect product of Dic6 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic6⋊C8
G = < a,b,c | a12=c8=1, b2=a6, bab-1=a-1, cac-1=a7, bc=cb >

Subgroups: 184 in 102 conjugacy classes, 61 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4×C8, C4⋊C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8×Q8, C4×C3⋊C8, C8×Dic3, Dic3⋊C8, C3×C4⋊C8, C4×Dic6, Dic6⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Q8, C23, D6, C2×C8, C22×C4, C2×Q8, C4○D4, C4×S3, C22×S3, C4×Q8, C22×C8, C8○D4, S3×C8, S3×C2×C4, D42S3, S3×Q8, C8×Q8, Dic6⋊C4, S3×C2×C8, D12.C4, Dic6⋊C8

Smallest permutation representation of Dic6⋊C8
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)
(1 123 7 129)(2 122 8 128)(3 121 9 127)(4 132 10 126)(5 131 11 125)(6 130 12 124)(13 102 19 108)(14 101 20 107)(15 100 21 106)(16 99 22 105)(17 98 23 104)(18 97 24 103)(25 118 31 112)(26 117 32 111)(27 116 33 110)(28 115 34 109)(29 114 35 120)(30 113 36 119)(37 188 43 182)(38 187 44 181)(39 186 45 192)(40 185 46 191)(41 184 47 190)(42 183 48 189)(49 71 55 65)(50 70 56 64)(51 69 57 63)(52 68 58 62)(53 67 59 61)(54 66 60 72)(73 154 79 148)(74 153 80 147)(75 152 81 146)(76 151 82 145)(77 150 83 156)(78 149 84 155)(85 159 91 165)(86 158 92 164)(87 157 93 163)(88 168 94 162)(89 167 95 161)(90 166 96 160)(133 171 139 177)(134 170 140 176)(135 169 141 175)(136 180 142 174)(137 179 143 173)(138 178 144 172)
(1 28 172 153 97 38 60 94)(2 35 173 148 98 45 49 89)(3 30 174 155 99 40 50 96)(4 25 175 150 100 47 51 91)(5 32 176 145 101 42 52 86)(6 27 177 152 102 37 53 93)(7 34 178 147 103 44 54 88)(8 29 179 154 104 39 55 95)(9 36 180 149 105 46 56 90)(10 31 169 156 106 41 57 85)(11 26 170 151 107 48 58 92)(12 33 171 146 108 43 59 87)(13 182 61 157 124 110 139 75)(14 189 62 164 125 117 140 82)(15 184 63 159 126 112 141 77)(16 191 64 166 127 119 142 84)(17 186 65 161 128 114 143 79)(18 181 66 168 129 109 144 74)(19 188 67 163 130 116 133 81)(20 183 68 158 131 111 134 76)(21 190 69 165 132 118 135 83)(22 185 70 160 121 113 136 78)(23 192 71 167 122 120 137 73)(24 187 72 162 123 115 138 80)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A 6B 6C 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 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 6 6 6 8 ··· 8 8 ··· 8 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 1 1 1 1 2 2 2 2 6 ··· 6 2 2 2 2 ··· 2 3 ··· 3 6 6 6 6 2 2 2 2 4 4 4 4 4 ··· 4

60 irreducible representations

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

Matrix representation of Dic6⋊C8 in GL4(𝔽73) generated by

 66 67 0 0 57 7 0 0 0 0 72 1 0 0 72 0
,
 0 51 0 0 10 0 0 0 0 0 13 62 0 0 2 60
,
 0 1 0 0 46 0 0 0 0 0 22 0 0 0 0 22
G:=sub<GL(4,GF(73))| [66,57,0,0,67,7,0,0,0,0,72,72,0,0,1,0],[0,10,0,0,51,0,0,0,0,0,13,2,0,0,62,60],[0,46,0,0,1,0,0,0,0,0,22,0,0,0,0,22] >;

Dic6⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes C_8
% in TeX

G:=Group("Dic6:C8");
// GroupNames label

G:=SmallGroup(192,389);
// by ID

G=gap.SmallGroup(192,389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,135,142,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=c^8=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations

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