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## G = Q8⋊5Dic6order 192 = 26·3

### 2nd semidirect product of Q8 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8⋊5Dic6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12⋊2Q8 — Q8⋊5Dic6
 Lower central C3 — C6 — C2×C12 — Q8⋊5Dic6
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q85Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 232 in 96 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4.Q16, C12⋊C8, C6.Q16, Q82Dic3, C122Q8, Q8×C12, Q85Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, Q16, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C3⋊Q16, C2×Dic6, C4○D12, C2×C3⋊D4, C4.Q16, C12.48D4, C2×C3⋊Q16, D4⋊D6, Q85Dic6

Smallest permutation representation of Q85Dic6
Regular action on 192 points
Generators in S192
(1 185 84 101)(2 186 73 102)(3 187 74 103)(4 188 75 104)(5 189 76 105)(6 190 77 106)(7 191 78 107)(8 192 79 108)(9 181 80 97)(10 182 81 98)(11 183 82 99)(12 184 83 100)(13 136 34 111)(14 137 35 112)(15 138 36 113)(16 139 25 114)(17 140 26 115)(18 141 27 116)(19 142 28 117)(20 143 29 118)(21 144 30 119)(22 133 31 120)(23 134 32 109)(24 135 33 110)(37 165 171 50)(38 166 172 51)(39 167 173 52)(40 168 174 53)(41 157 175 54)(42 158 176 55)(43 159 177 56)(44 160 178 57)(45 161 179 58)(46 162 180 59)(47 163 169 60)(48 164 170 49)(61 124 153 85)(62 125 154 86)(63 126 155 87)(64 127 156 88)(65 128 145 89)(66 129 146 90)(67 130 147 91)(68 131 148 92)(69 132 149 93)(70 121 150 94)(71 122 151 95)(72 123 152 96)
(1 137 84 112)(2 113 73 138)(3 139 74 114)(4 115 75 140)(5 141 76 116)(6 117 77 142)(7 143 78 118)(8 119 79 144)(9 133 80 120)(10 109 81 134)(11 135 82 110)(12 111 83 136)(13 184 34 100)(14 101 35 185)(15 186 36 102)(16 103 25 187)(17 188 26 104)(18 105 27 189)(19 190 28 106)(20 107 29 191)(21 192 30 108)(22 97 31 181)(23 182 32 98)(24 99 33 183)(37 126 171 87)(38 88 172 127)(39 128 173 89)(40 90 174 129)(41 130 175 91)(42 92 176 131)(43 132 177 93)(44 94 178 121)(45 122 179 95)(46 96 180 123)(47 124 169 85)(48 86 170 125)(49 62 164 154)(50 155 165 63)(51 64 166 156)(52 145 167 65)(53 66 168 146)(54 147 157 67)(55 68 158 148)(56 149 159 69)(57 70 160 150)(58 151 161 71)(59 72 162 152)(60 153 163 61)
(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 50 7 56)(2 49 8 55)(3 60 9 54)(4 59 10 53)(5 58 11 52)(6 57 12 51)(13 156 19 150)(14 155 20 149)(15 154 21 148)(16 153 22 147)(17 152 23 146)(18 151 24 145)(25 61 31 67)(26 72 32 66)(27 71 33 65)(28 70 34 64)(29 69 35 63)(30 68 36 62)(37 107 43 101)(38 106 44 100)(39 105 45 99)(40 104 46 98)(41 103 47 97)(42 102 48 108)(73 164 79 158)(74 163 80 157)(75 162 81 168)(76 161 82 167)(77 160 83 166)(78 159 84 165)(85 120 91 114)(86 119 92 113)(87 118 93 112)(88 117 94 111)(89 116 95 110)(90 115 96 109)(121 136 127 142)(122 135 128 141)(123 134 129 140)(124 133 130 139)(125 144 131 138)(126 143 132 137)(169 181 175 187)(170 192 176 186)(171 191 177 185)(172 190 178 184)(173 189 179 183)(174 188 180 182)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - - + - + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 Q16 C4○D4 C3⋊D4 Dic6 C4○D12 C8⋊C22 C3⋊Q16 D4⋊D6 kernel Q8⋊5Dic6 C12⋊C8 C6.Q16 Q8⋊2Dic3 C12⋊2Q8 Q8×C12 C4×Q8 C2×C12 C3×Q8 C42 C4⋊C4 C2×Q8 C12 C12 C2×C4 Q8 C4 C6 C4 C2 # reps 1 1 2 2 1 1 1 2 2 1 1 1 4 2 4 4 4 1 2 2

Matrix representation of Q85Dic6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 2 0 0 72 1
,
 1 0 0 0 0 1 0 0 0 0 1 16 0 0 9 72
,
 14 7 0 0 66 7 0 0 0 0 72 2 0 0 72 1
,
 48 37 0 0 62 25 0 0 0 0 36 33 0 0 16 37
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,72,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,9,0,0,16,72],[14,66,0,0,7,7,0,0,0,0,72,72,0,0,2,1],[48,62,0,0,37,25,0,0,0,0,36,16,0,0,33,37] >;

Q85Dic6 in GAP, Magma, Sage, TeX

Q_8\rtimes_5{\rm Dic}_6
% in TeX

G:=Group("Q8:5Dic6");
// GroupNames label

G:=SmallGroup(192,580);
// by ID

G=gap.SmallGroup(192,580);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,253,120,254,268,1123,297,136,6278]);
// Polycyclic

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

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