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G = Q8.5Dic6order 192 = 26·3

The non-split extension by Q8 of 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 — C12.6Q8 — Q8.5Dic6
 Lower central C3 — C6 — C2×C12 — Q8.5Dic6
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q8.5Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=a2c6, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 200 in 90 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×7], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8, Dic3 [×2], C12 [×2], C12 [×5], C2×C6, C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×Q8, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, Q8.Q8, C12⋊C8, C6.Q16, C12.Q8, Q82Dic3 [×2], C12.6Q8, Q8×C12, Q8.5Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C4○D8, C8.C22, C2×Dic6, C4○D12, C2×C3⋊D4, Q8.Q8, C12.48D4, Q8.11D6, Q8.13D6, Q8.5Dic6

Smallest permutation representation of Q8.5Dic6
Regular action on 192 points
Generators in S192
(1 91 162 144)(2 92 163 133)(3 93 164 134)(4 94 165 135)(5 95 166 136)(6 96 167 137)(7 85 168 138)(8 86 157 139)(9 87 158 140)(10 88 159 141)(11 89 160 142)(12 90 161 143)(13 189 48 55)(14 190 37 56)(15 191 38 57)(16 192 39 58)(17 181 40 59)(18 182 41 60)(19 183 42 49)(20 184 43 50)(21 185 44 51)(22 186 45 52)(23 187 46 53)(24 188 47 54)(25 107 73 118)(26 108 74 119)(27 97 75 120)(28 98 76 109)(29 99 77 110)(30 100 78 111)(31 101 79 112)(32 102 80 113)(33 103 81 114)(34 104 82 115)(35 105 83 116)(36 106 84 117)(61 180 132 148)(62 169 121 149)(63 170 122 150)(64 171 123 151)(65 172 124 152)(66 173 125 153)(67 174 126 154)(68 175 127 155)(69 176 128 156)(70 177 129 145)(71 178 130 146)(72 179 131 147)
(1 101 162 112)(2 102 163 113)(3 103 164 114)(4 104 165 115)(5 105 166 116)(6 106 167 117)(7 107 168 118)(8 108 157 119)(9 97 158 120)(10 98 159 109)(11 99 160 110)(12 100 161 111)(13 151 48 171)(14 152 37 172)(15 153 38 173)(16 154 39 174)(17 155 40 175)(18 156 41 176)(19 145 42 177)(20 146 43 178)(21 147 44 179)(22 148 45 180)(23 149 46 169)(24 150 47 170)(25 138 73 85)(26 139 74 86)(27 140 75 87)(28 141 76 88)(29 142 77 89)(30 143 78 90)(31 144 79 91)(32 133 80 92)(33 134 81 93)(34 135 82 94)(35 136 83 95)(36 137 84 96)(49 70 183 129)(50 71 184 130)(51 72 185 131)(52 61 186 132)(53 62 187 121)(54 63 188 122)(55 64 189 123)(56 65 190 124)(57 66 191 125)(58 67 192 126)(59 68 181 127)(60 69 182 128)
(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)
(1 187 168 59)(2 52 157 192)(3 185 158 57)(4 50 159 190)(5 183 160 55)(6 60 161 188)(7 181 162 53)(8 58 163 186)(9 191 164 51)(10 56 165 184)(11 189 166 49)(12 54 167 182)(13 136 42 89)(14 94 43 141)(15 134 44 87)(16 92 45 139)(17 144 46 85)(18 90 47 137)(19 142 48 95)(20 88 37 135)(21 140 38 93)(22 86 39 133)(23 138 40 91)(24 96 41 143)(25 68 79 121)(26 126 80 61)(27 66 81 131)(28 124 82 71)(29 64 83 129)(30 122 84 69)(31 62 73 127)(32 132 74 67)(33 72 75 125)(34 130 76 65)(35 70 77 123)(36 128 78 63)(97 153 114 179)(98 172 115 146)(99 151 116 177)(100 170 117 156)(101 149 118 175)(102 180 119 154)(103 147 120 173)(104 178 109 152)(105 145 110 171)(106 176 111 150)(107 155 112 169)(108 174 113 148)

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D8 C4○D12 C8.C22 Q8.11D6 Q8.13D6 kernel Q8.5Dic6 C12⋊C8 C6.Q16 C12.Q8 Q8⋊2Dic3 C12.6Q8 Q8×C12 C4×Q8 C2×C12 C3×Q8 C42 C4⋊C4 C2×Q8 C12 C2×C4 Q8 C6 C4 C6 C2 C2 # reps 1 1 1 1 2 1 1 1 2 2 1 1 1 2 4 4 4 4 1 2 2

Matrix representation of Q8.5Dic6 in GL4(𝔽73) generated by

 72 3 0 0 48 1 0 0 0 0 72 0 0 0 0 72
,
 36 25 0 0 62 37 0 0 0 0 43 13 0 0 60 30
,
 46 0 0 0 0 46 0 0 0 0 14 7 0 0 66 7
,
 52 64 0 0 57 21 0 0 0 0 8 47 0 0 39 65
G:=sub<GL(4,GF(73))| [72,48,0,0,3,1,0,0,0,0,72,0,0,0,0,72],[36,62,0,0,25,37,0,0,0,0,43,60,0,0,13,30],[46,0,0,0,0,46,0,0,0,0,14,66,0,0,7,7],[52,57,0,0,64,21,0,0,0,0,8,39,0,0,47,65] >;

Q8.5Dic6 in GAP, Magma, Sage, TeX

Q_8._5{\rm Dic}_6
% in TeX

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

G:=SmallGroup(192,581);
// by ID

G=gap.SmallGroup(192,581);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,184,1123,297,136,6278]);
// Polycyclic

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

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