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G = Q8⋊D20order 320 = 26·5

The semidirect product of Q8 and D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q81D20, C2013SD16, C42.57D10, (C4×Q8)⋊3D5, (C5×Q8)⋊8D4, C43(Q8⋊D5), (Q8×C20)⋊3C2, C203C826C2, C53(C4⋊SD16), C20.20(C2×D4), (C2×C20).66D4, C4.16(C2×D20), C4⋊C4.253D10, C204D4.7C2, D206C432C2, C4.12(C4○D20), C20.60(C4○D4), (C4×C20).98C22, (C2×Q8).160D10, C10.70(C2×SD16), C2.15(C207D4), C10.67(C4⋊D4), (C2×C20).347C23, (C2×D20).99C22, C2.10(D4⋊D10), C10.112(C8⋊C22), (Q8×C10).195C22, (C2×Q8⋊D5)⋊7C2, C2.7(C2×Q8⋊D5), (C2×C10).478(C2×D4), (C2×C4).249(C5⋊D4), (C5×C4⋊C4).284C22, (C2×C4).447(C22×D5), C22.155(C2×C5⋊D4), (C2×C52C8).101C22, SmallGroup(320,654)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Q8⋊D20
C1C5C10C20C2×C20C2×D20C204D4 — Q8⋊D20
C5C10C2×C20 — Q8⋊D20
C1C22C42C4×Q8

Generators and relations for Q8⋊D20
 G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Subgroups: 646 in 128 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], Q8, C23 [×2], D5 [×2], C10 [×3], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C2×D4 [×4], C2×Q8, C20 [×2], C20 [×2], C20 [×4], D10 [×6], C2×C10, D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C52C8 [×2], D20 [×8], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5 [×2], C4⋊SD16, C2×C52C8 [×2], Q8⋊D5 [×4], C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×D20 [×2], C2×D20 [×2], Q8×C10, C203C8, D206C4 [×2], C204D4, C2×Q8⋊D5 [×2], Q8×C20, Q8⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8⋊C22, D20 [×2], C5⋊D4 [×2], C22×D5, C4⋊SD16, Q8⋊D5 [×2], C2×D20, C4○D20, C2×C5⋊D4, C207D4, C2×Q8⋊D5, D4⋊D10, Q8⋊D20

Smallest permutation representation of Q8⋊D20
On 160 points
Generators in S160
(1 55 131 119)(2 56 132 120)(3 57 133 101)(4 58 134 102)(5 59 135 103)(6 60 136 104)(7 41 137 105)(8 42 138 106)(9 43 139 107)(10 44 140 108)(11 45 121 109)(12 46 122 110)(13 47 123 111)(14 48 124 112)(15 49 125 113)(16 50 126 114)(17 51 127 115)(18 52 128 116)(19 53 129 117)(20 54 130 118)(21 88 146 77)(22 89 147 78)(23 90 148 79)(24 91 149 80)(25 92 150 61)(26 93 151 62)(27 94 152 63)(28 95 153 64)(29 96 154 65)(30 97 155 66)(31 98 156 67)(32 99 157 68)(33 100 158 69)(34 81 159 70)(35 82 160 71)(36 83 141 72)(37 84 142 73)(38 85 143 74)(39 86 144 75)(40 87 145 76)
(1 28 131 153)(2 29 132 154)(3 30 133 155)(4 31 134 156)(5 32 135 157)(6 33 136 158)(7 34 137 159)(8 35 138 160)(9 36 139 141)(10 37 140 142)(11 38 121 143)(12 39 122 144)(13 40 123 145)(14 21 124 146)(15 22 125 147)(16 23 126 148)(17 24 127 149)(18 25 128 150)(19 26 129 151)(20 27 130 152)(41 70 105 81)(42 71 106 82)(43 72 107 83)(44 73 108 84)(45 74 109 85)(46 75 110 86)(47 76 111 87)(48 77 112 88)(49 78 113 89)(50 79 114 90)(51 80 115 91)(52 61 116 92)(53 62 117 93)(54 63 118 94)(55 64 119 95)(56 65 120 96)(57 66 101 97)(58 67 102 98)(59 68 103 99)(60 69 104 100)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 81)(22 100)(23 99)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(49 104)(50 103)(51 102)(52 101)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 155)(62 154)(63 153)(64 152)(65 151)(66 150)(67 149)(68 148)(69 147)(70 146)(71 145)(72 144)(73 143)(74 142)(75 141)(76 160)(77 159)(78 158)(79 157)(80 156)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4I5A5B8A8B8C8D10A···10F20A···20H20I···20AF
order12222244444···455888810···1020···2020···20
size1111404022224···422202020202···22···24···4

59 irreducible representations

dim11111122222222222444
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10D10C5⋊D4D20C4○D20C8⋊C22Q8⋊D5D4⋊D10
kernelQ8⋊D20C203C8D206C4C204D4C2×Q8⋊D5Q8×C20C2×C20C5×Q8C4×Q8C20C20C42C4⋊C4C2×Q8C2×C4Q8C4C10C4C2
# reps11212122242222888144

Matrix representation of Q8⋊D20 in GL4(𝔽41) generated by

1000
0100
00139
00140
,
1000
0100
001130
002630
,
283900
21600
00400
00040
,
283900
21300
00400
00401
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,11,26,0,0,30,30],[28,2,0,0,39,16,0,0,0,0,40,0,0,0,0,40],[28,2,0,0,39,13,0,0,0,0,40,40,0,0,0,1] >;

Q8⋊D20 in GAP, Magma, Sage, TeX

Q_8\rtimes D_{20}
% in TeX

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

G:=SmallGroup(320,654);
// by ID

G=gap.SmallGroup(320,654);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,184,1123,297,136,12550]);
// 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,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽