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G = Q86D20order 320 = 26·5

2nd semidirect product of Q8 and D20 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q86D20, C42.129D10, C10.1102+ 1+4, (C4×Q8)⋊10D5, (C5×Q8)⋊11D4, (C4×D20)⋊39C2, (Q8×C20)⋊12C2, C52(Q86D4), C4.26(C2×D20), C20.58(C2×D4), C2017(C4○D4), C204D413C2, C4⋊D2018C2, C4⋊C4.296D10, C43(Q82D5), (C2×Q8).205D10, C2.22(C22×D20), C10.20(C22×D4), (C2×C10).121C24, (C2×C20).170C23, (C4×C20).173C22, (C2×D20).30C22, C2.22(D48D10), C4⋊Dic5.399C22, (Q8×C10).221C22, (C22×D5).46C23, C22.142(C23×D5), (C2×Dic5).225C23, D10⋊C4.101C22, (C2×Q82D5)⋊4C2, (C2×C4×D5).82C22, C10.112(C2×C4○D4), C2.11(C2×Q82D5), (C5×C4⋊C4).349C22, (C2×C4).734(C22×D5), SmallGroup(320,1249)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Q86D20
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q82D5 — Q86D20
C5C2×C10 — Q86D20
C1C22C4×Q8

Generators and relations for Q86D20
 G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1366 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C5, C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], D5 [×6], C10 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×8], C20 [×3], D10 [×18], C2×C10, C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], C4×D5 [×12], D20 [×24], C2×Dic5 [×2], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×6], Q86D4, C4⋊Dic5, D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×6], C2×D20 [×15], Q82D5 [×8], Q8×C10, C4×D20 [×3], C204D4 [×3], C4⋊D20 [×6], Q8×C20, C2×Q82D5 [×2], Q86D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, D20 [×4], C22×D5 [×7], Q86D4, C2×D20 [×6], Q82D5 [×2], C23×D5, C22×D20, C2×Q82D5, D48D10, Q86D20

Smallest permutation representation of Q86D20
On 160 points
Generators in S160
(1 152 99 125)(2 153 100 126)(3 154 81 127)(4 155 82 128)(5 156 83 129)(6 157 84 130)(7 158 85 131)(8 159 86 132)(9 160 87 133)(10 141 88 134)(11 142 89 135)(12 143 90 136)(13 144 91 137)(14 145 92 138)(15 146 93 139)(16 147 94 140)(17 148 95 121)(18 149 96 122)(19 150 97 123)(20 151 98 124)(21 119 68 60)(22 120 69 41)(23 101 70 42)(24 102 71 43)(25 103 72 44)(26 104 73 45)(27 105 74 46)(28 106 75 47)(29 107 76 48)(30 108 77 49)(31 109 78 50)(32 110 79 51)(33 111 80 52)(34 112 61 53)(35 113 62 54)(36 114 63 55)(37 115 64 56)(38 116 65 57)(39 117 66 58)(40 118 67 59)
(1 74 99 27)(2 75 100 28)(3 76 81 29)(4 77 82 30)(5 78 83 31)(6 79 84 32)(7 80 85 33)(8 61 86 34)(9 62 87 35)(10 63 88 36)(11 64 89 37)(12 65 90 38)(13 66 91 39)(14 67 92 40)(15 68 93 21)(16 69 94 22)(17 70 95 23)(18 71 96 24)(19 72 97 25)(20 73 98 26)(41 147 120 140)(42 148 101 121)(43 149 102 122)(44 150 103 123)(45 151 104 124)(46 152 105 125)(47 153 106 126)(48 154 107 127)(49 155 108 128)(50 156 109 129)(51 157 110 130)(52 158 111 131)(53 159 112 132)(54 160 113 133)(55 141 114 134)(56 142 115 135)(57 143 116 136)(58 144 117 137)(59 145 118 138)(60 146 119 139)
(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)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(81 159)(82 158)(83 157)(84 156)(85 155)(86 154)(87 153)(88 152)(89 151)(90 150)(91 149)(92 148)(93 147)(94 146)(95 145)(96 144)(97 143)(98 142)(99 141)(100 160)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I4J4K4L4M4N4O5A5B10A···10F20A···20H20I···20AF
order12222···24···444444445510···1020···2020···20
size111120···202···244410101010222···22···24···4

65 irreducible representations

dim1111112222222444
type+++++++++++++++
imageC1C2C2C2C2C2D4D5C4○D4D10D10D10D202+ 1+4Q82D5D48D10
kernelQ86D20C4×D20C204D4C4⋊D20Q8×C20C2×Q82D5C5×Q8C4×Q8C20C42C4⋊C4C2×Q8Q8C10C4C2
# reps13361242466216144

Matrix representation of Q86D20 in GL4(𝔽41) generated by

1000
0100
004021
00371
,
1000
0100
003225
0009
,
273000
113200
0010
0001
,
302700
321100
004021
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,37,0,0,21,1],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,25,9],[27,11,0,0,30,32,0,0,0,0,1,0,0,0,0,1],[30,32,0,0,27,11,0,0,0,0,40,0,0,0,21,1] >;

Q86D20 in GAP, Magma, Sage, TeX

Q_8\rtimes_6D_{20}
% in TeX

G:=Group("Q8:6D20");
// GroupNames label

G:=SmallGroup(320,1249);
// by ID

G=gap.SmallGroup(320,1249);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,675,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
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