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G = D20.Q8order 320 = 26·5

1st non-split extension by D20 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.1Q8, C4.4(Q8×D5), C54(D4.Q8), C4.Q812D5, C20.16(C2×Q8), C4⋊C4.165D10, (C2×C8).142D10, C4.Dic106C2, D208C4.6C2, D206C4.6C2, C10.58(C4○D8), C4.77(C4○D20), C10.D818C2, C20.8Q831C2, (C2×Dic5).53D4, C22.222(D4×D5), D205C4.14C2, C20.169(C4○D4), C2.25(D40⋊C2), C10.74(C8⋊C22), (C2×C40).289C22, (C2×C20).287C23, (C2×D20).83C22, C10.38(C22⋊Q8), C2.15(D10⋊Q8), C4⋊Dic5.115C22, (C4×Dic5).39C22, C2.25(SD163D5), (C5×C4.Q8)⋊20C2, (C2×C10).292(C2×D4), (C5×C4⋊C4).80C22, (C2×C52C8).64C22, (C2×C4).390(C22×D5), SmallGroup(320,498)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.Q8
C1C5C10C2×C10C2×C20C2×D20D208C4 — D20.Q8
C5C10C2×C20 — D20.Q8
C1C22C2×C4C4.Q8

Generators and relations for D20.Q8
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=c3 >

Subgroups: 454 in 102 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×D20, C10.D8, D206C4, C20.8Q8, D205C4, C5×C4.Q8, C4.Dic10, D208C4, D20.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, C22×D5, D4.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D40⋊C2, SD163D5, D20.Q8

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120202244810102040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++-+++++-++
imageC1C2C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C4○D20C8⋊C22Q8×D5D4×D5D40⋊C2SD163D5
kernelD20.Q8C10.D8D206C4C20.8Q8D205C4C5×C4.Q8C4.Dic10D208C4D20C2×Dic5C4.Q8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111112222424812244

Matrix representation of D20.Q8 in GL6(𝔽41)

4000000
0400000
000100
00403400
00002132
00004020
,
4000000
4010000
000100
001000
0000400
000091
,
1390000
0400000
0040000
0004000
00002812
0000152
,
900000
090000
0040000
0004000
000090
0000132

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,21,40,0,0,0,0,32,20],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,28,15,0,0,0,0,12,2],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32] >;

D20.Q8 in GAP, Magma, Sage, TeX

D_{20}.Q_8
% in TeX

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

G:=SmallGroup(320,498);
// by ID

G=gap.SmallGroup(320,498);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,268,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

׿
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