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G = C82D20order 320 = 26·5

2nd semidirect product of C8 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C82D20, C407D4, C4.Q83D5, C52(C82D4), (C2×D40)⋊24C2, C4⋊D206C2, C4⋊C4.39D10, C4.51(C2×D20), (C2×C8).61D10, C20.131(C2×D4), D206C416C2, C20.30(C4○D4), C4.4(Q82D5), (C2×Dic5).50D4, (C22×D5).32D4, C22.217(D4×D5), C2.22(D40⋊C2), C10.44(C4⋊D4), C2.17(C4⋊D20), C10.70(C8⋊C22), (C2×C20).281C23, (C2×C40).110C22, (C2×D20).79C22, (C5×C4.Q8)⋊3C2, (C2×C8⋊D5)⋊2C2, (C2×C4×D5).37C22, (C2×C10).286(C2×D4), (C5×C4⋊C4).74C22, (C2×C52C8).58C22, (C2×C4).384(C22×D5), SmallGroup(320,492)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C82D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C82D20
C5C10C2×C20 — C82D20
C1C22C2×C4C4.Q8

Generators and relations for C82D20
 G = < a,b,c | a8=b20=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >

Subgroups: 742 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C52C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C82D4, C8⋊D5, D40, C2×C52C8, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, D206C4, C5×C4.Q8, C4⋊D20, C2×C8⋊D5, C2×D40, C82D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, D20, C22×D5, C82D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D40⋊C2, C82D20

Smallest permutation representation of C82D20
On 160 points
Generators in S160
(1 33 134 102 153 89 54 61)(2 103 55 34 154 62 135 90)(3 35 136 104 155 91 56 63)(4 105 57 36 156 64 137 92)(5 37 138 106 157 93 58 65)(6 107 59 38 158 66 139 94)(7 39 140 108 159 95 60 67)(8 109 41 40 160 68 121 96)(9 21 122 110 141 97 42 69)(10 111 43 22 142 70 123 98)(11 23 124 112 143 99 44 71)(12 113 45 24 144 72 125 100)(13 25 126 114 145 81 46 73)(14 115 47 26 146 74 127 82)(15 27 128 116 147 83 48 75)(16 117 49 28 148 76 129 84)(17 29 130 118 149 85 50 77)(18 119 51 30 150 78 131 86)(19 31 132 120 151 87 52 79)(20 101 53 32 152 80 133 88)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(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)(41 121)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(81 104)(82 103)(83 102)(84 101)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444455888810···102020202020···2040···40
size1111204040228820224420202···244448···84···4

44 irreducible representations

dim111111222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D20C8⋊C22Q82D5D4×D5D40⋊C2
kernelC82D20D206C4C5×C4.Q8C4⋊D20C2×C8⋊D5C2×D40C40C2×Dic5C22×D5C4.Q8C20C4⋊C4C2×C8C8C10C4C22C2
# reps121211211224282228

Matrix representation of C82D20 in GL6(𝔽41)

4000000
0400000
009153226
002632159
00915915
0026322632
,
010000
4000000
00357135
0034260
00135634
0060739
,
4000000
010000
0040700
000100
0000134
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,26,9,26,0,0,15,32,15,32,0,0,32,15,9,26,0,0,26,9,15,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,35,34,1,6,0,0,7,2,35,0,0,0,1,6,6,7,0,0,35,0,34,39],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,34,40] >;

C82D20 in GAP, Magma, Sage, TeX

C_8\rtimes_2D_{20}
% in TeX

G:=Group("C8:2D20");
// GroupNames label

G:=SmallGroup(320,492);
// by ID

G=gap.SmallGroup(320,492);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,58,438,102,12550]);
// Polycyclic

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

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