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G = D206Q8order 320 = 26·5

4th semidirect product of D20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D206Q8, C20.18D8, C42.81D10, C4⋊Q85D5, C4.12(Q8×D5), C4⋊C4.83D10, C55(D4⋊Q8), C203C835C2, C10.60(C2×D8), C20.39(C2×Q8), C4.16(D4⋊D5), (C4×D20).19C2, (C2×C20).157D4, C20.82(C4○D4), C10.D843C2, D206C4.15C2, (C4×C20).134C22, (C2×C20).405C23, C4.35(Q82D5), C10.76(C22⋊Q8), C2.13(D103Q8), (C2×D20).257C22, C10.97(C8.C22), C4⋊Dic5.348C22, C2.18(C20.C23), (C5×C4⋊Q8)⋊5C2, C2.15(C2×D4⋊D5), (C2×C10).536(C2×D4), (C2×C4).189(C5⋊D4), (C5×C4⋊C4).130C22, (C2×C4).502(C22×D5), C22.208(C2×C5⋊D4), (C2×C52C8).138C22, SmallGroup(320,714)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D206Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D206Q8
C5C10C2×C20 — D206Q8
C1C22C42C4⋊Q8

Generators and relations for D206Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 438 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C52C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4⋊Q8, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C203C8, C10.D8, D206C4, C4×D20, C5×C4⋊Q8, D206Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, C5⋊D4, C22×D5, D4⋊Q8, D4⋊D5, Q8×D5, Q82D5, C2×C5⋊D4, C2×D4⋊D5, C20.C23, D103Q8, D206Q8

Smallest permutation representation of D206Q8
On 160 points
Generators in S160
(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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 140)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 158)(42 157)(43 156)(44 155)(45 154)(46 153)(47 152)(48 151)(49 150)(50 149)(51 148)(52 147)(53 146)(54 145)(55 144)(56 143)(57 142)(58 141)(59 160)(60 159)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)
(1 112 140 98)(2 103 121 89)(3 114 122 100)(4 105 123 91)(5 116 124 82)(6 107 125 93)(7 118 126 84)(8 109 127 95)(9 120 128 86)(10 111 129 97)(11 102 130 88)(12 113 131 99)(13 104 132 90)(14 115 133 81)(15 106 134 92)(16 117 135 83)(17 108 136 94)(18 119 137 85)(19 110 138 96)(20 101 139 87)(21 47 68 148)(22 58 69 159)(23 49 70 150)(24 60 71 141)(25 51 72 152)(26 42 73 143)(27 53 74 154)(28 44 75 145)(29 55 76 156)(30 46 77 147)(31 57 78 158)(32 48 79 149)(33 59 80 160)(34 50 61 151)(35 41 62 142)(36 52 63 153)(37 43 64 144)(38 54 65 155)(39 45 66 146)(40 56 67 157)
(1 31 140 78)(2 32 121 79)(3 33 122 80)(4 34 123 61)(5 35 124 62)(6 36 125 63)(7 37 126 64)(8 38 127 65)(9 39 128 66)(10 40 129 67)(11 21 130 68)(12 22 131 69)(13 23 132 70)(14 24 133 71)(15 25 134 72)(16 26 135 73)(17 27 136 74)(18 28 137 75)(19 29 138 76)(20 30 139 77)(41 116 142 82)(42 117 143 83)(43 118 144 84)(44 119 145 85)(45 120 146 86)(46 101 147 87)(47 102 148 88)(48 103 149 89)(49 104 150 90)(50 105 151 91)(51 106 152 92)(52 107 153 93)(53 108 154 94)(54 109 155 95)(55 110 156 96)(56 111 157 97)(57 112 158 98)(58 113 159 99)(59 114 160 100)(60 115 141 81)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12222244444444455888810···1020···2020···20
size111120202222488202022202020202···24···48···8

47 irreducible representations

dim1111112222222244444
type++++++-+++++-+-+
imageC1C2C2C2C2C2Q8D4D5D8C4○D4D10D10C5⋊D4C8.C22D4⋊D5Q8×D5Q82D5C20.C23
kernelD206Q8C203C8C10.D8D206C4C4×D20C5×C4⋊Q8D20C2×C20C4⋊Q8C20C20C42C4⋊C4C2×C4C10C4C4C4C2
# reps1122112224224814224

Matrix representation of D206Q8 in GL6(𝔽41)

010000
4060000
0013700
00214000
0000400
0000040
,
010000
100000
0013700
0004000
0000123
0000040
,
100000
010000
0017700
00352400
000090
0000132
,
100000
010000
001000
000100
00003239
000009

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,21,0,0,0,0,37,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,37,40,0,0,0,0,0,0,1,0,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,35,0,0,0,0,7,24,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,39,9] >;

D206Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_6Q_8
% in TeX

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

G:=SmallGroup(320,714);
// by ID

G=gap.SmallGroup(320,714);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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