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G = C85D20order 320 = 26·5

2nd semidirect product of C8 and D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C85D20, C4023D4, C207SD16, C42.260D10, (C4×C8)⋊12D5, (C4×C40)⋊17C2, C51(C85D4), C41(C40⋊C2), C202Q81C2, C4.30(C2×D20), (C2×C4).79D20, (C2×C20).376D4, C204D4.2C2, C20.273(C2×D4), (C2×C8).317D10, C10.4(C2×SD16), C10.3(C41D4), C2.5(C204D4), (C2×D20).3C22, C22.89(C2×D20), (C2×C40).389C22, (C4×C20).306C22, (C2×C20).722C23, (C2×Dic10).4C22, (C2×C40⋊C2)⋊7C2, C2.7(C2×C40⋊C2), (C2×C10).105(C2×D4), (C2×C4).665(C22×D5), SmallGroup(320,320)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C85D20
C1C5C10C20C2×C20C2×D20C204D4 — C85D20
C5C10C2×C20 — C85D20
C1C22C42C4×C8

Generators and relations for C85D20
 G = < a,b,c | a8=b20=c2=1, ab=ba, cac=a3, cbc=b-1 >

Subgroups: 782 in 142 conjugacy classes, 55 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×8], C2×D4 [×4], C2×Q8 [×2], Dic5 [×2], C20 [×6], D10 [×6], C2×C10, C4×C8, C41D4, C4⋊Q8, C2×SD16 [×4], C40 [×4], Dic10 [×4], D20 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C85D4, C40⋊C2 [×8], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×D20 [×2], C2×D20 [×2], C4×C40, C202Q8, C204D4, C2×C40⋊C2 [×4], C85D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×4], C2×D4 [×3], D10 [×3], C41D4, C2×SD16 [×2], D20 [×6], C22×D5, C85D4, C40⋊C2 [×4], C2×D20 [×3], C204D4, C2×C40⋊C2 [×2], C85D20

Smallest permutation representation of C85D20
On 160 points
Generators in S160
(1 30 88 80 146 103 41 124)(2 31 89 61 147 104 42 125)(3 32 90 62 148 105 43 126)(4 33 91 63 149 106 44 127)(5 34 92 64 150 107 45 128)(6 35 93 65 151 108 46 129)(7 36 94 66 152 109 47 130)(8 37 95 67 153 110 48 131)(9 38 96 68 154 111 49 132)(10 39 97 69 155 112 50 133)(11 40 98 70 156 113 51 134)(12 21 99 71 157 114 52 135)(13 22 100 72 158 115 53 136)(14 23 81 73 159 116 54 137)(15 24 82 74 160 117 55 138)(16 25 83 75 141 118 56 139)(17 26 84 76 142 119 57 140)(18 27 85 77 143 120 58 121)(19 28 86 78 144 101 59 122)(20 29 87 79 145 102 60 123)
(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 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(101 125)(102 124)(103 123)(104 122)(105 121)(106 140)(107 139)(108 138)(109 137)(110 136)(111 135)(112 134)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(141 150)(142 149)(143 148)(144 147)(145 146)(151 160)(152 159)(153 158)(154 157)(155 156)

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

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

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

86 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F20A···20X40A···40AF
order1222224···444558···810···1020···2040···40
size111140402···24040222···22···22···22···2

86 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2D4D4D5SD16D10D10D20D20C40⋊C2
kernelC85D20C4×C40C202Q8C204D4C2×C40⋊C2C40C2×C20C4×C8C20C42C2×C8C8C2×C4C4
# reps1111442282416832

Matrix representation of C85D20 in GL4(𝔽41) generated by

393400
271300
001410
00216
,
40000
04000
00272
002511
,
40000
8100
00302
002211
G:=sub<GL(4,GF(41))| [39,27,0,0,34,13,0,0,0,0,14,2,0,0,10,16],[40,0,0,0,0,40,0,0,0,0,27,25,0,0,2,11],[40,8,0,0,0,1,0,0,0,0,30,22,0,0,2,11] >;

C85D20 in GAP, Magma, Sage, TeX

C_8\rtimes_5D_{20}
% in TeX

G:=Group("C8:5D20");
// GroupNames label

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

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

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

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