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G = Q85D20order 320 = 26·5

1st semidirect product of Q8 and D20 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q85D20, C42.128D10, C10.112- 1+4, (C4×Q8)⋊9D5, (C5×Q8)⋊10D4, (C4×D20)⋊38C2, (Q8×C20)⋊11C2, C52(Q85D4), C20.57(C2×D4), C4.25(C2×D20), C4⋊D2017C2, C4⋊C4.295D10, D1013(C4○D4), D102Q818C2, C4.D2020C2, (C2×Q8).204D10, C2.21(C22×D20), C10.19(C22×D4), (C4×C20).172C22, (C2×C20).498C23, (C2×C10).120C24, D10⋊C4.6C22, (C2×D20).224C22, C4⋊Dic5.306C22, (Q8×C10).220C22, (C2×Dic5).54C23, (C22×D5).45C23, C22.141(C23×D5), C2.12(Q8.10D10), (C2×Dic10).155C22, (C2×Q8×D5)⋊4C2, C2.29(D5×C4○D4), (C2×Q82D5)⋊3C2, (C2×C4×D5).81C22, C10.145(C2×C4○D4), (C5×C4⋊C4).348C22, (C2×C4).168(C22×D5), SmallGroup(320,1248)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Q85D20
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — Q85D20
C5C2×C10 — Q85D20
C1C22C4×Q8

Generators and relations for Q85D20
 G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1126 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×13], C5, C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×4], Q8 [×6], C23 [×4], D5 [×5], C10 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], C20 [×6], C20 [×4], D10 [×2], D10 [×11], C2×C10, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×6], C4×D5 [×12], D20 [×12], C2×Dic5, C2×Dic5 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×3], Q85D4, C4⋊Dic5 [×3], D10⋊C4, D10⋊C4 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×6], C2×D20 [×6], Q8×D5 [×4], Q82D5 [×4], Q8×C10, C4×D20 [×3], C4.D20 [×3], C4⋊D20 [×3], D102Q8 [×3], Q8×C20, C2×Q8×D5, C2×Q82D5, Q85D20
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], Q85D4, C2×D20 [×6], C23×D5, C22×D20, Q8.10D10, D5×C4○D4, Q85D20

Smallest permutation representation of Q85D20
On 160 points
Generators in S160
(1 142 35 97)(2 143 36 98)(3 144 37 99)(4 145 38 100)(5 146 39 81)(6 147 40 82)(7 148 21 83)(8 149 22 84)(9 150 23 85)(10 151 24 86)(11 152 25 87)(12 153 26 88)(13 154 27 89)(14 155 28 90)(15 156 29 91)(16 157 30 92)(17 158 31 93)(18 159 32 94)(19 160 33 95)(20 141 34 96)(41 117 78 121)(42 118 79 122)(43 119 80 123)(44 120 61 124)(45 101 62 125)(46 102 63 126)(47 103 64 127)(48 104 65 128)(49 105 66 129)(50 106 67 130)(51 107 68 131)(52 108 69 132)(53 109 70 133)(54 110 71 134)(55 111 72 135)(56 112 73 136)(57 113 74 137)(58 114 75 138)(59 115 76 139)(60 116 77 140)
(1 59 35 76)(2 77 36 60)(3 41 37 78)(4 79 38 42)(5 43 39 80)(6 61 40 44)(7 45 21 62)(8 63 22 46)(9 47 23 64)(10 65 24 48)(11 49 25 66)(12 67 26 50)(13 51 27 68)(14 69 28 52)(15 53 29 70)(16 71 30 54)(17 55 31 72)(18 73 32 56)(19 57 33 74)(20 75 34 58)(81 119 146 123)(82 124 147 120)(83 101 148 125)(84 126 149 102)(85 103 150 127)(86 128 151 104)(87 105 152 129)(88 130 153 106)(89 107 154 131)(90 132 155 108)(91 109 156 133)(92 134 157 110)(93 111 158 135)(94 136 159 112)(95 113 160 137)(96 138 141 114)(97 115 142 139)(98 140 143 116)(99 117 144 121)(100 122 145 118)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 80)(60 79)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 140)(119 139)(120 138)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I4J4K4L4M4N4O4P5A5B10A···10F20A···20H20I···20AF
order1222222224···4444444445510···1020···2020···20
size111110102020202···24441010202020222···22···24···4

65 irreducible representations

dim111111112222222444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D202- 1+4Q8.10D10D5×C4○D4
kernelQ85D20C4×D20C4.D20C4⋊D20D102Q8Q8×C20C2×Q8×D5C2×Q82D5C5×Q8C4×Q8D10C42C4⋊C4C2×Q8Q8C10C2C2
# reps1333311142466216144

Matrix representation of Q85D20 in GL6(𝔽41)

4000000
0400000
0040000
0004000
000090
00002832
,
4000000
0400000
001000
000100
00002937
00002612
,
120000
40400000
0040100
0053500
000010
00003540
,
100000
40400000
0040000
005100
000010
00003540

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,28,0,0,0,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,26,0,0,0,0,37,12],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,35,0,0,0,0,0,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,1,35,0,0,0,0,0,40] >;

Q85D20 in GAP, Magma, Sage, TeX

Q_8\rtimes_5D_{20}
% in TeX

G:=Group("Q8:5D20");
// GroupNames label

G:=SmallGroup(320,1248);
// by ID

G=gap.SmallGroup(320,1248);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,12550]);
// Polycyclic

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

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