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G = C20.57D8order 320 = 26·5

11st non-split extension by C20 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.57D8, C20.41SD16, C42.194D10, C20.21M4(2), D4⋊(C52C8), C55(D4⋊C8), (C5×D4)⋊3C8, C203C89C2, (C4×D4).1D5, C10.26C4≀C2, C20.31(C2×C8), (D4×C20).1C2, C4⋊C4.3Dic5, C4.30(D4⋊D5), (D4×C10).20C4, (C2×C20).489D4, (C2×D4).4Dic5, (C4×C20).45C22, C4.16(D4.D5), C4.1(C4.Dic5), C10.30(C22⋊C8), C2.2(D4⋊Dic5), C10.37(D4⋊C4), C2.2(D42Dic5), C2.5(C20.55D4), C22.29(C23.D5), (C4×C52C8)⋊2C2, C4.1(C2×C52C8), (C5×C4⋊C4).18C4, (C2×C20).232(C2×C4), (C2×C4).37(C2×Dic5), (C2×C4).161(C5⋊D4), (C2×C10).156(C22⋊C4), SmallGroup(320,92)

Series: Derived Chief Lower central Upper central

C1C20 — C20.57D8
C1C5C10C2×C10C2×C20C4×C20C203C8 — C20.57D8
C5C10C20 — C20.57D8
C1C2×C4C42C4×D4

Generators and relations for C20.57D8
 G = < a,b,c | a20=b8=1, c2=a5, bab-1=cac-1=a9, cbc-1=a5b-1 >

Subgroups: 214 in 82 conjugacy classes, 39 normal (35 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, C20 [×4], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, C4⋊C8, C4×D4, C52C8 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4⋊C8, C2×C52C8 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4×C52C8, C203C8, D4×C20, C20.57D8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D8, SD16, Dic5 [×2], D10, C22⋊C8, D4⋊C4, C4≀C2, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], D4⋊C8, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, C23.D5, C20.55D4, D4⋊Dic5, D42Dic5, C20.57D8

Smallest permutation representation of C20.57D8
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 24 92 48 104 122 152 66)(2 33 93 57 105 131 153 75)(3 22 94 46 106 140 154 64)(4 31 95 55 107 129 155 73)(5 40 96 44 108 138 156 62)(6 29 97 53 109 127 157 71)(7 38 98 42 110 136 158 80)(8 27 99 51 111 125 159 69)(9 36 100 60 112 134 160 78)(10 25 81 49 113 123 141 67)(11 34 82 58 114 132 142 76)(12 23 83 47 115 121 143 65)(13 32 84 56 116 130 144 74)(14 21 85 45 117 139 145 63)(15 30 86 54 118 128 146 72)(16 39 87 43 119 137 147 61)(17 28 88 52 120 126 148 70)(18 37 89 41 101 135 149 79)(19 26 90 50 102 124 150 68)(20 35 91 59 103 133 151 77)
(1 71 6 76 11 61 16 66)(2 80 7 65 12 70 17 75)(3 69 8 74 13 79 18 64)(4 78 9 63 14 68 19 73)(5 67 10 72 15 77 20 62)(21 155 26 160 31 145 36 150)(22 144 27 149 32 154 37 159)(23 153 28 158 33 143 38 148)(24 142 29 147 34 152 39 157)(25 151 30 156 35 141 40 146)(41 101 46 106 51 111 56 116)(42 110 47 115 52 120 57 105)(43 119 48 104 53 109 58 114)(44 108 49 113 54 118 59 103)(45 117 50 102 55 107 60 112)(81 138 86 123 91 128 96 133)(82 127 87 132 92 137 97 122)(83 136 88 121 93 126 98 131)(84 125 89 130 94 135 99 140)(85 134 90 139 95 124 100 129)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L10A···10F10G···10N20A···20H20I···20X
order1222224444444444558···8888810···1010···1020···2020···20
size11114411112222442210···10202020202···24···42···24···4

68 irreducible representations

dim1111111222222222222444
type++++++++--+-
imageC1C2C2C2C4C4C8D4D5M4(2)D8SD16D10Dic5Dic5C4≀C2C5⋊D4C52C8C4.Dic5D4⋊D5D4.D5D42Dic5
kernelC20.57D8C4×C52C8C203C8D4×C20C5×C4⋊C4D4×C10C5×D4C2×C20C4×D4C20C20C20C42C4⋊C4C2×D4C10C2×C4D4C4C4C4C2
# reps1111228222222224888224

Matrix representation of C20.57D8 in GL6(𝔽41)

3200000
0320000
0031000
001400
000090
000009
,
3710000
3440000
00301000
00371100
0000317
0000280
,
3710000
1640000
00301000
00371100
0000317
00002810

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,31,1,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[37,34,0,0,0,0,1,4,0,0,0,0,0,0,30,37,0,0,0,0,10,11,0,0,0,0,0,0,31,28,0,0,0,0,7,0],[37,16,0,0,0,0,1,4,0,0,0,0,0,0,30,37,0,0,0,0,10,11,0,0,0,0,0,0,31,28,0,0,0,0,7,10] >;

C20.57D8 in GAP, Magma, Sage, TeX

C_{20}._{57}D_8
% in TeX

G:=Group("C20.57D8");
// GroupNames label

G:=SmallGroup(320,92);
// by ID

G=gap.SmallGroup(320,92);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,1123,570,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;
// generators/relations

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