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G = Q85Dic6order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q85Dic6, C12.23Q16, C42.54D6, (C3×Q8)⋊4Q8, C4⋊C4.249D6, C34(C4.Q16), (C2×C12).64D4, (Q8×C12).3C2, (C4×Q8).10S3, C12.29(C2×Q8), C6.34(C2×Q16), (C2×Q8).180D6, C12⋊C8.15C2, C4.13(C2×Dic6), C12.56(C4○D4), C4.63(C4○D12), C2.9(D4⋊D6), (C4×C12).92C22, C122Q8.14C2, C6.Q16.10C2, C4.11(C3⋊Q16), Q82Dic3.8C2, C6.65(C22⋊Q8), C6.110(C8⋊C22), (C2×C12).343C23, (C6×Q8).191C22, C4⋊Dic3.140C22, C2.16(C12.48D4), C2.6(C2×C3⋊Q16), (C2×C6).474(C2×D4), (C2×C3⋊C8).98C22, (C2×C4).248(C3⋊D4), (C3×C4⋊C4).280C22, (C2×C4).443(C22×S3), C22.153(C2×C3⋊D4), SmallGroup(192,580)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Q85Dic6
C1C3C6C12C2×C12C4⋊Dic3C122Q8 — Q85Dic6
C3C6C2×C12 — Q85Dic6
C1C22C42C4×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 [×3], C3, C4 [×2], C4 [×2], C4 [×6], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], C2×C6, C42, C42, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, Q8⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4.Q16, C12⋊C8, C6.Q16 [×2], Q82Dic3 [×2], C122Q8, Q8×C12, Q85Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C3⋊Q16 [×2], 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 94 113 158)(2 95 114 159)(3 96 115 160)(4 85 116 161)(5 86 117 162)(6 87 118 163)(7 88 119 164)(8 89 120 165)(9 90 109 166)(10 91 110 167)(11 92 111 168)(12 93 112 157)(13 151 31 99)(14 152 32 100)(15 153 33 101)(16 154 34 102)(17 155 35 103)(18 156 36 104)(19 145 25 105)(20 146 26 106)(21 147 27 107)(22 148 28 108)(23 149 29 97)(24 150 30 98)(37 142 73 130)(38 143 74 131)(39 144 75 132)(40 133 76 121)(41 134 77 122)(42 135 78 123)(43 136 79 124)(44 137 80 125)(45 138 81 126)(46 139 82 127)(47 140 83 128)(48 141 84 129)(49 178 68 186)(50 179 69 187)(51 180 70 188)(52 169 71 189)(53 170 72 190)(54 171 61 191)(55 172 62 192)(56 173 63 181)(57 174 64 182)(58 175 65 183)(59 176 66 184)(60 177 67 185)
(1 143 113 131)(2 132 114 144)(3 133 115 121)(4 122 116 134)(5 135 117 123)(6 124 118 136)(7 137 119 125)(8 126 120 138)(9 139 109 127)(10 128 110 140)(11 141 111 129)(12 130 112 142)(13 54 31 61)(14 62 32 55)(15 56 33 63)(16 64 34 57)(17 58 35 65)(18 66 36 59)(19 60 25 67)(20 68 26 49)(21 50 27 69)(22 70 28 51)(23 52 29 71)(24 72 30 53)(37 93 73 157)(38 158 74 94)(39 95 75 159)(40 160 76 96)(41 85 77 161)(42 162 78 86)(43 87 79 163)(44 164 80 88)(45 89 81 165)(46 166 82 90)(47 91 83 167)(48 168 84 92)(97 169 149 189)(98 190 150 170)(99 171 151 191)(100 192 152 172)(101 173 153 181)(102 182 154 174)(103 175 155 183)(104 184 156 176)(105 177 145 185)(106 186 146 178)(107 179 147 187)(108 188 148 180)
(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)
(1 102 7 108)(2 101 8 107)(3 100 9 106)(4 99 10 105)(5 98 11 104)(6 97 12 103)(13 167 19 161)(14 166 20 160)(15 165 21 159)(16 164 22 158)(17 163 23 157)(18 162 24 168)(25 85 31 91)(26 96 32 90)(27 95 33 89)(28 94 34 88)(29 93 35 87)(30 92 36 86)(37 183 43 189)(38 182 44 188)(39 181 45 187)(40 192 46 186)(41 191 47 185)(42 190 48 184)(49 121 55 127)(50 132 56 126)(51 131 57 125)(52 130 58 124)(53 129 59 123)(54 128 60 122)(61 140 67 134)(62 139 68 133)(63 138 69 144)(64 137 70 143)(65 136 71 142)(66 135 72 141)(73 175 79 169)(74 174 80 180)(75 173 81 179)(76 172 82 178)(77 171 83 177)(78 170 84 176)(109 146 115 152)(110 145 116 151)(111 156 117 150)(112 155 118 149)(113 154 119 148)(114 153 120 147)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E···4I4J4K6A6B6C8A8B8C8D12A12B12C12D12E···12P
order1222344444···44466688881212121212···12
size1111222224···424242221212121222224···4

39 irreducible representations

dim11111122222222222444
type++++++++-+++--+-+
imageC1C2C2C2C2C2S3D4Q8D6D6D6Q16C4○D4C3⋊D4Dic6C4○D12C8⋊C22C3⋊Q16D4⋊D6
kernelQ85Dic6C12⋊C8C6.Q16Q82Dic3C122Q8Q8×C12C4×Q8C2×C12C3×Q8C42C4⋊C4C2×Q8C12C12C2×C4Q8C4C6C4C2
# reps11221112211142444122

Matrix representation of Q85Dic6 in GL4(𝔽73) generated by

1000
0100
00722
00721
,
1000
0100
00116
00972
,
14700
66700
00722
00721
,
483700
622500
003633
001637
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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