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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 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8, C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5 [×3], C10, C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×4], Dic5, C20 [×2], C20 [×2], D10 [×9], C2×C10, D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C52C8, C40 [×2], C4×D5 [×2], D20 [×8], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], C82D4, C8⋊D5 [×2], D40 [×2], C2×C52C8, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×D20 [×2], C2×D20 [×2], D206C4 [×2], C5×C4.Q8, C4⋊D20 [×2], C2×C8⋊D5, C2×D40, C82D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], D20 [×2], C22×D5, C82D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D40⋊C2 [×2], C82D20

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

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

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

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

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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