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G = Q84Dic6order 192 = 26·3

1st semidirect product of Q8 and Dic6 acting via Dic6/C12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q84Dic6, C42.53D6, C12.48SD16, (C3×Q8)⋊3Q8, (C4×Q8).9S3, C4⋊C4.248D6, C34(Q8⋊Q8), (C2×C12).63D4, (Q8×C12).2C2, C12.28(C2×Q8), (C2×Q8).179D6, C12⋊C8.14C2, C4.12(C2×Dic6), C6.68(C2×SD16), C12.55(C4○D4), C4.62(C4○D12), (C4×C12).91C22, C122Q8.13C2, Q82Dic3.7C2, C6.64(C22⋊Q8), (C2×C12).342C23, C4.13(Q82S3), C2.9(Q8.14D6), C12.Q8.10C2, (C6×Q8).190C22, C6.110(C8.C22), C4⋊Dic3.139C22, C2.15(C12.48D4), (C2×C6).473(C2×D4), (C2×C3⋊C8).97C22, C2.6(C2×Q82S3), (C2×C4).247(C3⋊D4), (C3×C4⋊C4).279C22, (C2×C4).442(C22×S3), C22.152(C2×C3⋊D4), SmallGroup(192,579)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Q84Dic6
C1C3C6C12C2×C12C4⋊Dic3C122Q8 — Q84Dic6
C3C6C2×C12 — Q84Dic6
C1C22C42C4×Q8

Generators and relations for Q84Dic6
 G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, 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, C4.Q8 [×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, Q8⋊Q8, C12⋊C8, C12.Q8 [×2], Q82Dic3 [×2], C122Q8, Q8×C12, Q84Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×SD16, C8.C22, Q82S3 [×2], C2×Dic6, C4○D12, C2×C3⋊D4, Q8⋊Q8, C12.48D4, C2×Q82S3, Q8.14D6, Q84Dic6

Smallest permutation representation of Q84Dic6
Regular action on 192 points
Generators in S192
(1 117 89 51)(2 118 90 52)(3 119 91 53)(4 120 92 54)(5 109 93 55)(6 110 94 56)(7 111 95 57)(8 112 96 58)(9 113 85 59)(10 114 86 60)(11 115 87 49)(12 116 88 50)(13 166 99 28)(14 167 100 29)(15 168 101 30)(16 157 102 31)(17 158 103 32)(18 159 104 33)(19 160 105 34)(20 161 106 35)(21 162 107 36)(22 163 108 25)(23 164 97 26)(24 165 98 27)(37 137 80 68)(38 138 81 69)(39 139 82 70)(40 140 83 71)(41 141 84 72)(42 142 73 61)(43 143 74 62)(44 144 75 63)(45 133 76 64)(46 134 77 65)(47 135 78 66)(48 136 79 67)(121 180 181 151)(122 169 182 152)(123 170 183 153)(124 171 184 154)(125 172 185 155)(126 173 186 156)(127 174 187 145)(128 175 188 146)(129 176 189 147)(130 177 190 148)(131 178 191 149)(132 179 192 150)
(1 36 89 162)(2 25 90 163)(3 26 91 164)(4 27 92 165)(5 28 93 166)(6 29 94 167)(7 30 95 168)(8 31 96 157)(9 32 85 158)(10 33 86 159)(11 34 87 160)(12 35 88 161)(13 109 99 55)(14 110 100 56)(15 111 101 57)(16 112 102 58)(17 113 103 59)(18 114 104 60)(19 115 105 49)(20 116 106 50)(21 117 107 51)(22 118 108 52)(23 119 97 53)(24 120 98 54)(37 177 80 148)(38 178 81 149)(39 179 82 150)(40 180 83 151)(41 169 84 152)(42 170 73 153)(43 171 74 154)(44 172 75 155)(45 173 76 156)(46 174 77 145)(47 175 78 146)(48 176 79 147)(61 183 142 123)(62 184 143 124)(63 185 144 125)(64 186 133 126)(65 187 134 127)(66 188 135 128)(67 189 136 129)(68 190 137 130)(69 191 138 131)(70 192 139 132)(71 181 140 121)(72 182 141 122)
(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)
(1 183 7 189)(2 182 8 188)(3 181 9 187)(4 192 10 186)(5 191 11 185)(6 190 12 184)(13 138 19 144)(14 137 20 143)(15 136 21 142)(16 135 22 141)(17 134 23 140)(18 133 24 139)(25 84 31 78)(26 83 32 77)(27 82 33 76)(28 81 34 75)(29 80 35 74)(30 79 36 73)(37 161 43 167)(38 160 44 166)(39 159 45 165)(40 158 46 164)(41 157 47 163)(42 168 48 162)(49 155 55 149)(50 154 56 148)(51 153 57 147)(52 152 58 146)(53 151 59 145)(54 150 60 156)(61 101 67 107)(62 100 68 106)(63 99 69 105)(64 98 70 104)(65 97 71 103)(66 108 72 102)(85 127 91 121)(86 126 92 132)(87 125 93 131)(88 124 94 130)(89 123 95 129)(90 122 96 128)(109 178 115 172)(110 177 116 171)(111 176 117 170)(112 175 118 169)(113 174 119 180)(114 173 120 179)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim11111122222222222444
type++++++++-+++--+-
imageC1C2C2C2C2C2S3D4Q8D6D6D6SD16C4○D4C3⋊D4Dic6C4○D12C8.C22Q82S3Q8.14D6
kernelQ84Dic6C12⋊C8C12.Q8Q82Dic3C122Q8Q8×C12C4×Q8C2×C12C3×Q8C42C4⋊C4C2×Q8C12C12C2×C4Q8C4C6C4C2
# reps11221112211142444122

Matrix representation of Q84Dic6 in GL6(𝔽73)

100000
010000
0014800
0037200
0000720
0000072
,
7200000
0720000
00126900
00186100
00004360
00001330
,
130000
48720000
001000
000100
0000721
0000720
,
69130000
3840000
0072000
0070100
0000260
00006271

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,18,0,0,0,0,69,61,0,0,0,0,0,0,43,13,0,0,0,0,60,30],[1,48,0,0,0,0,3,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[69,38,0,0,0,0,13,4,0,0,0,0,0,0,72,70,0,0,0,0,0,1,0,0,0,0,0,0,2,62,0,0,0,0,60,71] >;

Q84Dic6 in GAP, Magma, Sage, TeX

Q_8\rtimes_4{\rm Dic}_6
% in TeX

G:=Group("Q8:4Dic6");
// GroupNames label

G:=SmallGroup(192,579);
// by ID

G=gap.SmallGroup(192,579);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,184,1123,297,136,6278]);
// 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,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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