Copied to
clipboard

G = C83D20order 320 = 26·5

3rd semidirect product of C8 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D20, C4010D4, C53(C8⋊D4), C2.D811D5, C4⋊C4.51D10, (C2×C8).66D10, C4.54(C2×D20), D102Q87C2, C20.134(C2×D4), D206C423C2, C20.41(C4○D4), C10.Q1620C2, C4⋊D20.10C2, (C2×Dic5).58D4, (C22×D5).37D4, C22.232(D4×D5), C2.24(D8⋊D5), C2.20(C4⋊D20), C10.47(C4⋊D4), C10.43(C8⋊C22), (C2×C20).302C23, (C2×C40).144C22, C4.10(Q82D5), (C2×D20).88C22, C2.23(Q16⋊D5), C10.71(C8.C22), (C2×Dic10).94C22, (C5×C2.D8)⋊8C2, (C2×C8⋊D5)⋊6C2, (C2×C40⋊C2)⋊22C2, (C2×C4×D5).45C22, (C2×C10).307(C2×D4), (C5×C4⋊C4).95C22, (C2×C52C8).72C22, (C2×C4).405(C22×D5), SmallGroup(320,513)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C83D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C83D20
C5C10C2×C20 — C83D20
C1C22C2×C4C2.D8

Generators and relations for C83D20
 G = < a,b,c | a8=b20=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >

Subgroups: 598 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C8⋊D4, C8⋊D5 [×2], C40⋊C2 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D206C4, C10.Q16, C5×C2.D8, C4⋊D20, D102Q8, C2×C8⋊D5, C2×C40⋊C2, C83D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, D20 [×2], C22×D5, C8⋊D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D8⋊D5, Q16⋊D5, C83D20

Smallest permutation representation of C83D20
On 160 points
Generators in S160
(1 131 115 96 25 67 60 146)(2 147 41 68 26 97 116 132)(3 133 117 98 27 69 42 148)(4 149 43 70 28 99 118 134)(5 135 119 100 29 71 44 150)(6 151 45 72 30 81 120 136)(7 137 101 82 31 73 46 152)(8 153 47 74 32 83 102 138)(9 139 103 84 33 75 48 154)(10 155 49 76 34 85 104 140)(11 121 105 86 35 77 50 156)(12 157 51 78 36 87 106 122)(13 123 107 88 37 79 52 158)(14 159 53 80 38 89 108 124)(15 125 109 90 39 61 54 160)(16 141 55 62 40 91 110 126)(17 127 111 92 21 63 56 142)(18 143 57 64 22 93 112 128)(19 129 113 94 23 65 58 144)(20 145 59 66 24 95 114 130)
(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 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 146)(62 145)(63 144)(64 143)(65 142)(66 141)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)(81 140)(82 139)(83 138)(84 137)(85 136)(86 135)(87 134)(88 133)(89 132)(90 131)(91 130)(92 129)(93 128)(94 127)(95 126)(96 125)(97 124)(98 123)(99 122)(100 121)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444455888810···102020202020···2040···40
size1111204022882040224420202···244448···84···4

44 irreducible representations

dim1111111122222222444444
type++++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D20C8⋊C22C8.C22Q82D5D4×D5D8⋊D5Q16⋊D5
kernelC83D20D206C4C10.Q16C5×C2.D8C4⋊D20D102Q8C2×C8⋊D5C2×C40⋊C2C40C2×Dic5C22×D5C2.D8C20C4⋊C4C2×C8C8C10C10C4C22C2C2
# reps1111111121122428112244

Matrix representation of C83D20 in GL8(𝔽41)

10000000
01000000
004000000
000400000
0000011526
00003218240
00002821320
00000382120
,
01000000
400000000
004010000
005350000
00001090
000000401
0000180400
00001840400
,
400000000
01000000
004000000
00510000
000040000
000023100
000023010
000000040

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,32,28,0,0,0,0,0,1,18,21,38,0,0,0,0,15,24,3,21,0,0,0,0,26,0,20,20],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,0,18,18,0,0,0,0,0,0,0,40,0,0,0,0,9,40,40,40,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,23,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;

C83D20 in GAP, Magma, Sage, TeX

C_8\rtimes_3D_{20}
% in TeX

G:=Group("C8:3D20");
// GroupNames label

G:=SmallGroup(320,513);
// by ID

G=gap.SmallGroup(320,513);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,219,58,438,102,12550]);
// Polycyclic

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

׿
×
𝔽