Copied to
clipboard

G = C87D20order 320 = 26·5

1st semidirect product of C8 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C87D20, C404D4, D102D8, C2.D83D5, C52(C87D4), C2.14(D5×D8), (C2×D40)⋊16C2, C4⋊D207C2, C4⋊C4.48D10, C4.53(C2×D20), C10.30(C2×D8), (C2×C8).231D10, C20.133(C2×D4), D206C421C2, C10.75(C4○D8), C20.38(C4○D4), (C2×C40).83C22, C4.9(Q82D5), (C22×D5).86D4, C22.229(D4×D5), C10.46(C4⋊D4), C2.19(C4⋊D20), (C2×C20).299C23, (C2×Dic5).148D4, (C2×D20).86C22, C2.13(Q8.D10), (D5×C2×C8)⋊2C2, (C5×C2.D8)⋊5C2, (C2×C10).304(C2×D4), (C5×C4⋊C4).92C22, (C2×C4×D5).306C22, (C2×C4).402(C22×D5), (C2×C52C8).243C22, SmallGroup(320,510)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C87D20
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C87D20
C5C10C2×C20 — C87D20
C1C22C2×C4C2.D8

Generators and relations for C87D20
 G = < a,b,c | a8=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 742 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C8 [×2], C8, C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5 [×4], C10 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×3], D8 [×2], C22×C4, C2×D4 [×4], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, C52C8, C40 [×2], C4×D5 [×2], D20 [×8], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], C87D4, 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×C2.D8, C4⋊D20 [×2], D5×C2×C8, C2×D40, C87D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, D20 [×2], C22×D5, C87D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D5×D8, Q8.D10, C87D20

Smallest permutation representation of C87D20
On 160 points
Generators in S160
(1 34 93 50 155 79 130 104)(2 105 131 80 156 51 94 35)(3 36 95 52 157 61 132 106)(4 107 133 62 158 53 96 37)(5 38 97 54 159 63 134 108)(6 109 135 64 160 55 98 39)(7 40 99 56 141 65 136 110)(8 111 137 66 142 57 100 21)(9 22 81 58 143 67 138 112)(10 113 139 68 144 59 82 23)(11 24 83 60 145 69 140 114)(12 115 121 70 146 41 84 25)(13 26 85 42 147 71 122 116)(14 117 123 72 148 43 86 27)(15 28 87 44 149 73 124 118)(16 119 125 74 150 45 88 29)(17 30 89 46 151 75 126 120)(18 101 127 76 152 47 90 31)(19 32 91 48 153 77 128 102)(20 103 129 78 154 49 92 33)
(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)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 62)(42 61)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(81 136)(82 135)(83 134)(84 133)(85 132)(86 131)(87 130)(88 129)(89 128)(90 127)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 140)(98 139)(99 138)(100 137)(141 143)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444558888888810···102020202020···2040···40
size11111010404022881010222222101010102···244448···84···4

50 irreducible representations

dim11111122222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D8D10D10C4○D8D20Q82D5D4×D5D5×D8Q8.D10
kernelC87D20D206C4C5×C2.D8C4⋊D20D5×C2×C8C2×D40C40C2×Dic5C22×D5C2.D8C20D10C4⋊C4C2×C8C10C8C4C22C2C2
# reps12121121122442482244

Matrix representation of C87D20 in GL4(𝔽41) generated by

40000
04000
00030
001517
,
323000
112700
00320
00409
,
04000
40000
004018
0001
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,15,0,0,30,17],[32,11,0,0,30,27,0,0,0,0,32,40,0,0,0,9],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,18,1] >;

C87D20 in GAP, Magma, Sage, TeX

C_8\rtimes_7D_{20}
% in TeX

G:=Group("C8:7D20");
// GroupNames label

G:=SmallGroup(320,510);
// by ID

G=gap.SmallGroup(320,510);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,438,102,12550]);
// Polycyclic

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

׿
×
𝔽