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G = D20.3Q8order 320 = 26·5

1st non-split extension by D20 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.3Q8, C42.38D10, C4⋊C87D5, C53(D4.Q8), C4.45(Q8×D5), C405C413C2, C406C417C2, (C2×C4).40D20, (C4×D20).13C2, (C2×C20).246D4, (C2×C8).132D10, C20.104(C2×Q8), D205C4.4C2, C10.14(C4○D8), (C2×C40).26C22, (C4×C20).73C22, C20.6Q810C2, C20.288(C4○D4), C2.19(C8⋊D10), C10.16(C8⋊C22), (C2×C20).757C23, C22.120(C2×D20), C10.32(C22⋊Q8), C4⋊Dic5.20C22, C4.112(D42D5), C2.16(D407C2), C2.13(D102Q8), (C2×D20).203C22, (C5×C4⋊C8)⋊9C2, (C2×C10).140(C2×D4), (C2×C4).702(C22×D5), SmallGroup(320,474)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.3Q8
C1C5C10C20C2×C20C2×D20C4×D20 — D20.3Q8
C5C10C2×C20 — D20.3Q8
C1C22C42C4⋊C8

Generators and relations for D20.3Q8
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5c2, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a15c3 >

Subgroups: 470 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], 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, C40 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, D4.Q8, C10.D4 [×2], C4⋊Dic5 [×3], D10⋊C4, C4×C20, C2×C40 [×2], C2×C4×D5, C2×D20, C406C4, C405C4, D205C4 [×2], C5×C4⋊C8, C20.6Q8, C4×D20, D20.3Q8
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, D20 [×2], C22×D5, D4.Q8, C2×D20, D42D5, Q8×D5, D102Q8, D407C2, C8⋊D10, D20.3Q8

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222244444444455888810···1020···2020···2040···40
size1111202022224202040402244442···22···24···44···4

59 irreducible representations

dim11111112222222224444
type+++++++-++++++--+
imageC1C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8D20D407C2C8⋊C22D42D5Q8×D5C8⋊D10
kernelD20.3Q8C406C4C405C4D205C4C5×C4⋊C8C20.6Q8C4×D20D20C2×C20C4⋊C8C20C42C2×C8C10C2×C4C2C10C4C4C2
# reps111211122222448161224

Matrix representation of D20.3Q8 in GL4(𝔽41) generated by

25200
391300
0010
0001
,
22500
133900
00400
00040
,
263100
10400
0001
00400
,
9000
0900
00352
0026
G:=sub<GL(4,GF(41))| [25,39,0,0,2,13,0,0,0,0,1,0,0,0,0,1],[2,13,0,0,25,39,0,0,0,0,40,0,0,0,0,40],[26,10,0,0,31,4,0,0,0,0,0,40,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,35,2,0,0,2,6] >;

D20.3Q8 in GAP, Magma, Sage, TeX

D_{20}._3Q_8
% in TeX

G:=Group("D20.3Q8");
// GroupNames label

G:=SmallGroup(320,474);
// by ID

G=gap.SmallGroup(320,474);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,142,1123,136,12550]);
// 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,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^15*c^3>;
// generators/relations

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