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G = C8.D20order 320 = 26·5

1st non-split extension by C8 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.1D4, C8.1D20, C42.21D10, C8⋊C44D5, C202Q84C2, (C2×C20).39D4, (C2×C4).28D20, C4.35(C2×D20), (C2×C8).56D10, C51(C8.2D4), C20.278(C2×D4), (C4×C20).6C22, (C2×Dic20)⋊10C2, C10.8(C41D4), (C2×C40).57C22, C4.D20.5C2, C2.10(C204D4), (C2×C20).736C23, C2.9(C8.D10), (C2×D20).11C22, C22.100(C2×D20), C10.5(C8.C22), (C2×Dic10).11C22, (C5×C8⋊C4)⋊5C2, (C2×C40⋊C2).2C2, (C2×C10).119(C2×D4), (C2×C4).680(C22×D5), SmallGroup(320,342)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.D20
C1C5C10C20C2×C20C2×D20C4.D20 — C8.D20
C5C10C2×C20 — C8.D20
C1C22C42C8⋊C4

Generators and relations for C8.D20
 G = < a,b,c | a8=b20=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 590 in 124 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, D5, C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8 [×3], Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C40 [×4], Dic10 [×6], D20 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×2], C22×D5, C8.2D4, C40⋊C2 [×4], Dic20 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×Dic10, C2×Dic10 [×2], C2×D20, C5×C8⋊C4, C202Q8, C4.D20, C2×C40⋊C2 [×2], C2×Dic20 [×2], C8.D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8.C22 [×2], D20 [×6], C22×D5, C8.2D4, C2×D20 [×3], C204D4, C8.D10 [×2], C8.D20

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222444444455888810···1020···2020···2040···40
size11114022444040402244442···22···24···44···4

56 irreducible representations

dim111111222222244
type+++++++++++++--
imageC1C2C2C2C2C2D4D4D5D10D10D20D20C8.C22C8.D10
kernelC8.D20C5×C8⋊C4C202Q8C4.D20C2×C40⋊C2C2×Dic20C40C2×C20C8⋊C4C42C2×C8C8C2×C4C10C2
# reps1111224222416828

Matrix representation of C8.D20 in GL6(𝔽41)

010000
4000000
0037214029
002013121
00150420
000152128
,
0400000
100000
001212137
0040548
0021282940
001330136
,
0400000
4000000
002940204
001122821
00204026
002821260

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,20,15,0,0,0,21,13,0,15,0,0,40,12,4,21,0,0,29,1,20,28],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,12,40,21,13,0,0,1,5,28,30,0,0,21,4,29,1,0,0,37,8,40,36],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,29,1,20,28,0,0,40,12,4,21,0,0,20,28,0,26,0,0,4,21,26,0] >;

C8.D20 in GAP, Magma, Sage, TeX

C_8.D_{20}
% in TeX

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

G:=SmallGroup(320,342);
// by ID

G=gap.SmallGroup(320,342);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations

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