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G = C20.50D8order 320 = 26·5

4th non-split extension by C20 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.50D8, D43Dic10, C42.44D10, (C5×D4)⋊3Q8, (C4×D4).2D5, C54(D4⋊Q8), (D4×C20).2C2, C203C817C2, C10.50(C2×D8), (C2×C20).57D4, C20.25(C2×Q8), C4.23(D4⋊D5), C4⋊C4.238D10, C202Q814C2, C4.9(C2×Dic10), (C2×D4).185D10, C20.45(C4○D4), C4.59(C4○D20), C10.D830C2, (C4×C20).78C22, D4⋊Dic5.7C2, (C2×C20).332C23, C10.61(C22⋊Q8), C2.7(D4.9D10), (D4×C10).227C22, C4⋊Dic5.136C22, C2.12(C20.48D4), C10.107(C8.C22), C2.6(C2×D4⋊D5), (C2×C10).463(C2×D4), (C2×C4).243(C5⋊D4), (C5×C4⋊C4).269C22, (C2×C52C8).89C22, (C2×C4).432(C22×D5), C22.146(C2×C5⋊D4), SmallGroup(320,634)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.50D8
C1C5C10C20C2×C20C4⋊Dic5C202Q8 — C20.50D8
C5C10C2×C20 — C20.50D8
C1C22C42C4×D4

Generators and relations for C20.50D8
 G = < a,b,c | a20=b8=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 358 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4⋊Q8, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C203C8, C10.D8 [×2], D4⋊Dic5 [×2], C202Q8, D4×C20, C20.50D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, D8 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×D8, C8.C22, Dic10 [×2], C5⋊D4 [×2], C22×D5, D4⋊Q8, D4⋊D5 [×2], C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×D4⋊D5, D4.9D10, C20.50D8

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222244444444455888810···1010···1020···2020···20
size1111442222444404022202020202···24···42···24···4

59 irreducible representations

dim11111122222222222444
type+++++++-+++++--+-
imageC1C2C2C2C2C2D4Q8D5D8C4○D4D10D10D10C5⋊D4Dic10C4○D20C8.C22D4⋊D5D4.9D10
kernelC20.50D8C203C8C10.D8D4⋊Dic5C202Q8D4×C20C2×C20C5×D4C4×D4C20C20C42C4⋊C4C2×D4C2×C4D4C4C10C4C2
# reps11221122242222888144

Matrix representation of C20.50D8 in GL4(𝔽41) generated by

40000
04000
002530
002739
,
01700
121700
00022
00130
,
01700
29000
00022
00130
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,25,27,0,0,30,39],[0,12,0,0,17,17,0,0,0,0,0,13,0,0,22,0],[0,29,0,0,17,0,0,0,0,0,0,13,0,0,22,0] >;

C20.50D8 in GAP, Magma, Sage, TeX

C_{20}._{50}D_8
% in TeX

G:=Group("C20.50D8");
// GroupNames label

G:=SmallGroup(320,634);
// by ID

G=gap.SmallGroup(320,634);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations

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