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## G = Dic6⋊9Q8order 192 = 26·3

### 7th semidirect product of Dic6 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊9Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C2×Dic6 — C4×Dic6 — Dic6⋊9Q8
 Lower central C3 — C2×C6 — Dic6⋊9Q8
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for Dic69Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, bc=cb, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 448 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, Q82, C4×Dic6, Dic6⋊C4, C12⋊Q8, Dic3⋊Q8, C3×C4⋊Q8, Dic69Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, 2+ 1+4, S3×Q8, S3×C23, Q82, D46D6, C2×S3×Q8, Dic69Q8

Smallest permutation representation of Dic69Q8
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)
(1 169 7 175)(2 180 8 174)(3 179 9 173)(4 178 10 172)(5 177 11 171)(6 176 12 170)(13 153 19 147)(14 152 20 146)(15 151 21 145)(16 150 22 156)(17 149 23 155)(18 148 24 154)(25 181 31 187)(26 192 32 186)(27 191 33 185)(28 190 34 184)(29 189 35 183)(30 188 36 182)(37 157 43 163)(38 168 44 162)(39 167 45 161)(40 166 46 160)(41 165 47 159)(42 164 48 158)(49 89 55 95)(50 88 56 94)(51 87 57 93)(52 86 58 92)(53 85 59 91)(54 96 60 90)(61 114 67 120)(62 113 68 119)(63 112 69 118)(64 111 70 117)(65 110 71 116)(66 109 72 115)(73 131 79 125)(74 130 80 124)(75 129 81 123)(76 128 82 122)(77 127 83 121)(78 126 84 132)(97 143 103 137)(98 142 104 136)(99 141 105 135)(100 140 106 134)(101 139 107 133)(102 138 108 144)
(1 122 192 157)(2 129 181 164)(3 124 182 159)(4 131 183 166)(5 126 184 161)(6 121 185 168)(7 128 186 163)(8 123 187 158)(9 130 188 165)(10 125 189 160)(11 132 190 167)(12 127 191 162)(13 102 71 58)(14 97 72 53)(15 104 61 60)(16 99 62 55)(17 106 63 50)(18 101 64 57)(19 108 65 52)(20 103 66 59)(21 98 67 54)(22 105 68 49)(23 100 69 56)(24 107 70 51)(25 42 174 75)(26 37 175 82)(27 44 176 77)(28 39 177 84)(29 46 178 79)(30 41 179 74)(31 48 180 81)(32 43 169 76)(33 38 170 83)(34 45 171 78)(35 40 172 73)(36 47 173 80)(85 152 143 115)(86 147 144 110)(87 154 133 117)(88 149 134 112)(89 156 135 119)(90 151 136 114)(91 146 137 109)(92 153 138 116)(93 148 139 111)(94 155 140 118)(95 150 141 113)(96 145 142 120)
(1 15 192 61)(2 16 181 62)(3 17 182 63)(4 18 183 64)(5 19 184 65)(6 20 185 66)(7 21 186 67)(8 22 187 68)(9 23 188 69)(10 24 189 70)(11 13 190 71)(12 14 191 72)(25 113 174 150)(26 114 175 151)(27 115 176 152)(28 116 177 153)(29 117 178 154)(30 118 179 155)(31 119 180 156)(32 120 169 145)(33 109 170 146)(34 110 171 147)(35 111 172 148)(36 112 173 149)(37 136 82 90)(38 137 83 91)(39 138 84 92)(40 139 73 93)(41 140 74 94)(42 141 75 95)(43 142 76 96)(44 143 77 85)(45 144 78 86)(46 133 79 87)(47 134 80 88)(48 135 81 89)(49 158 105 123)(50 159 106 124)(51 160 107 125)(52 161 108 126)(53 162 97 127)(54 163 98 128)(55 164 99 129)(56 165 100 130)(57 166 101 131)(58 167 102 132)(59 168 103 121)(60 157 104 122)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 2+ 1+4 S3×Q8 D4⋊6D6 kernel Dic6⋊9Q8 C4×Dic6 Dic6⋊C4 C12⋊Q8 Dic3⋊Q8 C3×C4⋊Q8 C4⋊Q8 Dic6 C42 C4⋊C4 C2×Q8 C6 C4 C2 # reps 1 2 4 4 4 1 1 8 1 4 2 1 4 2

Matrix representation of Dic69Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 2 10 0 0 0 0 6 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 11 8
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 2 5
,
 10 4 0 0 0 0 4 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 10 0 0 0 0 6 11

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,2,6,0,0,0,0,10,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,11,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,2,0,0,0,0,0,5],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,6,0,0,0,0,10,11] >;

Dic69Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_9Q_8
% in TeX

G:=Group("Dic6:9Q8");
// GroupNames label

G:=SmallGroup(192,1281);
// by ID

G=gap.SmallGroup(192,1281);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,570,185,192,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations

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