Copied to
clipboard

G = C8.20D20order 320 = 26·5

6th non-split extension by C8 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.20D20, C40.18D4, D20.20D4, Dic10.20D4, M4(2).9D10, C4.135(D4×D5), C8.C45D5, C4.56(C2×D20), (C2×C8).70D10, C20.136(C2×D4), C52(D4.5D4), (C2×Dic20)⋊21C2, C4.12D203C2, C8.D10.2C2, C10.49(C4⋊D4), C2.22(C4⋊D20), (C2×C20).312C23, (C2×C40).102C22, D20.3C4.2C2, C4○D20.39C22, C22.6(Q82D5), (C5×M4(2)).6C22, C4.Dic5.37C22, (C2×Dic10).98C22, (C5×C8.C4)⋊6C2, (C2×C10).3(C4○D4), (C2×C4).113(C22×D5), SmallGroup(320,523)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.20D20
C1C5C10C20C2×C20C4○D20D20.3C4 — C8.20D20
C5C10C2×C20 — C8.20D20
C1C2C2×C4C8.C4

Generators and relations for C8.20D20
 G = < a,b,c | a40=1, b4=c2=a20, bab-1=a31, cac-1=a-1, cbc-1=b3 >

Subgroups: 414 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D5, C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C52C8, C40 [×2], C40 [×2], Dic10, Dic10 [×4], C4×D5, D20, C2×Dic5 [×2], C5⋊D4, C2×C20, D4.5D4, C8×D5, C8⋊D5, C40⋊C2 [×2], Dic20 [×4], C4.Dic5, C2×C40, C5×M4(2) [×2], C2×Dic10 [×2], C4○D20, C4.12D20 [×2], C5×C8.C4, D20.3C4, C2×Dic20, C8.D10 [×2], C8.20D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.5D4, C2×D20, D4×D5, Q82D5, C4⋊D20, C8.20D20

Smallest permutation representation of C8.20D20
On 160 points
Generators in S160
(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)
(1 109 48 155 21 89 68 135)(2 100 49 146 22 120 69 126)(3 91 50 137 23 111 70 157)(4 82 51 128 24 102 71 148)(5 113 52 159 25 93 72 139)(6 104 53 150 26 84 73 130)(7 95 54 141 27 115 74 121)(8 86 55 132 28 106 75 152)(9 117 56 123 29 97 76 143)(10 108 57 154 30 88 77 134)(11 99 58 145 31 119 78 125)(12 90 59 136 32 110 79 156)(13 81 60 127 33 101 80 147)(14 112 61 158 34 92 41 138)(15 103 62 149 35 83 42 129)(16 94 63 140 36 114 43 160)(17 85 64 131 37 105 44 151)(18 116 65 122 38 96 45 142)(19 107 66 153 39 87 46 133)(20 98 67 144 40 118 47 124)
(1 155 21 135)(2 154 22 134)(3 153 23 133)(4 152 24 132)(5 151 25 131)(6 150 26 130)(7 149 27 129)(8 148 28 128)(9 147 29 127)(10 146 30 126)(11 145 31 125)(12 144 32 124)(13 143 33 123)(14 142 34 122)(15 141 35 121)(16 140 36 160)(17 139 37 159)(18 138 38 158)(19 137 39 157)(20 136 40 156)(41 116 61 96)(42 115 62 95)(43 114 63 94)(44 113 64 93)(45 112 65 92)(46 111 66 91)(47 110 67 90)(48 109 68 89)(49 108 69 88)(50 107 70 87)(51 106 71 86)(52 105 72 85)(53 104 73 84)(54 103 74 83)(55 102 75 82)(56 101 76 81)(57 100 77 120)(58 99 78 119)(59 98 79 118)(60 97 80 117)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222444445588888881010101020202020202040···4040···40
size11220222040402222488202022442222444···48···8

44 irreducible representations

dim111111222222224444
type+++++++++++++-++-
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D20D4.5D4D4×D5Q82D5C8.20D20
kernelC8.20D20C4.12D20C5×C8.C4D20.3C4C2×Dic20C8.D10C40Dic10D20C8.C4C2×C10C2×C8M4(2)C8C5C4C22C1
# reps121112211222482228

Matrix representation of C8.20D20 in GL4(𝔽41) generated by

2442737
1003110
363500
40352737
,
4253518
3420179
3413516
36281923
,
1762235
25302613
3413516
3437390
G:=sub<GL(4,GF(41))| [24,10,36,40,4,0,35,35,27,31,0,27,37,10,0,37],[4,34,34,36,25,20,1,28,35,17,35,19,18,9,16,23],[17,25,34,34,6,30,1,37,22,26,35,39,35,13,16,0] >;

C8.20D20 in GAP, Magma, Sage, TeX

C_8._{20}D_{20}
% in TeX

G:=Group("C8.20D20");
// GroupNames label

G:=SmallGroup(320,523);
// by ID

G=gap.SmallGroup(320,523);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,1123,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^4=c^2=a^20,b*a*b^-1=a^31,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

׿
×
𝔽