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## G = C8.6Dic6order 192 = 26·3

### 3rd non-split extension by C8 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8.6Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C8×Dic3 — C8.6Dic6
 Lower central C3 — C6 — C2×C12 — C8.6Dic6
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C8.6Dic6
G = < a,b,c | a8=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=a4b-1 >

Subgroups: 208 in 86 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×C8, C4.Q8, C2.D8, C2.D8, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C8.5Q8, C12.Q8, C8×Dic3, C241C4, C3×C2.D8, C4.Dic6, C8.6Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C4○D8, C2×Dic6, S3×D4, S3×Q8, C8.5Q8, C12⋊Q8, D83S3, D24⋊C2, C8.6Dic6

Smallest permutation representation of C8.6Dic6
Regular action on 192 points
Generators in S192
(1 117 64 102 51 92 77 125)(2 126 78 93 52 103 65 118)(3 119 66 104 53 94 79 127)(4 128 80 95 54 105 67 120)(5 109 68 106 55 96 81 129)(6 130 82 85 56 107 69 110)(7 111 70 108 57 86 83 131)(8 132 84 87 58 97 71 112)(9 113 72 98 59 88 73 121)(10 122 74 89 60 99 61 114)(11 115 62 100 49 90 75 123)(12 124 76 91 50 101 63 116)(13 180 30 164 45 151 189 135)(14 136 190 152 46 165 31 169)(15 170 32 166 47 153 191 137)(16 138 192 154 48 167 33 171)(17 172 34 168 37 155 181 139)(18 140 182 156 38 157 35 173)(19 174 36 158 39 145 183 141)(20 142 184 146 40 159 25 175)(21 176 26 160 41 147 185 143)(22 144 186 148 42 161 27 177)(23 178 28 162 43 149 187 133)(24 134 188 150 44 163 29 179)
(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)
(1 165 7 159)(2 135 8 141)(3 163 9 157)(4 133 10 139)(5 161 11 167)(6 143 12 137)(13 132 19 126)(14 108 20 102)(15 130 21 124)(16 106 22 100)(17 128 23 122)(18 104 24 98)(25 117 31 111)(26 91 32 85)(27 115 33 109)(28 89 34 95)(29 113 35 119)(30 87 36 93)(37 105 43 99)(38 127 44 121)(39 103 45 97)(40 125 46 131)(41 101 47 107)(42 123 48 129)(49 138 55 144)(50 166 56 160)(51 136 57 142)(52 164 58 158)(53 134 59 140)(54 162 60 168)(61 155 67 149)(62 171 68 177)(63 153 69 147)(64 169 70 175)(65 151 71 145)(66 179 72 173)(73 156 79 150)(74 172 80 178)(75 154 81 148)(76 170 82 176)(77 152 83 146)(78 180 84 174)(86 184 92 190)(88 182 94 188)(90 192 96 186)(110 185 116 191)(112 183 118 189)(114 181 120 187)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - - + + + - - + - + image C1 C2 C2 C2 C2 C2 S3 Q8 Q8 D4 D6 D6 Dic6 C4○D8 S3×Q8 S3×D4 D8⋊3S3 D24⋊C2 kernel C8.6Dic6 C12.Q8 C8×Dic3 C24⋊1C4 C3×C2.D8 C4.Dic6 C2.D8 C3⋊C8 C24 C2×Dic3 C4⋊C4 C2×C8 C8 C6 C4 C22 C2 C2 # reps 1 2 1 1 1 2 1 2 2 2 2 1 4 8 1 1 2 2

Matrix representation of C8.6Dic6 in GL6(𝔽73)

 19 52 0 0 0 0 52 54 0 0 0 0 0 0 57 57 0 0 0 0 16 57 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 14 30 0 0 0 0 30 59 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 19 52 0 0 0 0 52 54 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 18 50 0 0 0 0 68 55

G:=sub<GL(6,GF(73))| [19,52,0,0,0,0,52,54,0,0,0,0,0,0,57,16,0,0,0,0,57,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,14,30,0,0,0,0,30,59,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[19,52,0,0,0,0,52,54,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,18,68,0,0,0,0,50,55] >;

C8.6Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("C8.6Dic6");
// GroupNames label

G:=SmallGroup(192,437);
// by ID

G=gap.SmallGroup(192,437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,926,219,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^12=1,c^2=b^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b^-1>;
// generators/relations

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