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G = C4.5D40order 320 = 26·5

5th non-split extension by C4 of D40 acting via D40/C40=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.5D40, C20.30D8, C20.26SD16, C42.261D10, (C4×C8)⋊5D5, (C4×C40)⋊5C2, C2.5(C2×D40), C10.3(C2×D8), C202Q82C2, (C2×C4).80D20, C51(C4.4D8), D205C41C2, C204D4.3C2, (C2×C20).377D4, (C2×C8).287D10, C4.5(C40⋊C2), C10.5(C2×SD16), (C2×D20).4C22, C22.90(C2×D20), C4⋊Dic5.5C22, C20.218(C4○D4), C4.102(C4○D20), (C2×C20).723C23, (C2×C40).347C22, (C4×C20).307C22, C10.5(C4.4D4), C2.10(C4.D20), C2.8(C2×C40⋊C2), (C2×C10).106(C2×D4), (C2×C4).666(C22×D5), SmallGroup(320,321)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4.5D40
C1C5C10C20C2×C20C2×D20C204D4 — C4.5D40
C5C10C2×C20 — C4.5D40
C1C22C42C4×C8

Generators and relations for C4.5D40
 G = < a,b,c | a4=b40=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 686 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C4⋊C4 [×3], C2×C8 [×2], C2×D4 [×4], C2×Q8, Dic5 [×2], C20 [×6], D10 [×6], C2×C10, C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C40 [×2], Dic10 [×2], D20 [×8], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4.4D8, C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C2×D20 [×2], C2×D20 [×2], D205C4 [×4], C4×C40, C202Q8, C204D4, C4.5D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, D20 [×2], C22×D5, C4.4D8, C40⋊C2 [×2], D40 [×2], C2×D20, C4○D20 [×2], C4.D20, C2×C40⋊C2, C2×D40, C4.5D40

Smallest permutation representation of C4.5D40
On 160 points
Generators in S160
(1 82 41 146)(2 83 42 147)(3 84 43 148)(4 85 44 149)(5 86 45 150)(6 87 46 151)(7 88 47 152)(8 89 48 153)(9 90 49 154)(10 91 50 155)(11 92 51 156)(12 93 52 157)(13 94 53 158)(14 95 54 159)(15 96 55 160)(16 97 56 121)(17 98 57 122)(18 99 58 123)(19 100 59 124)(20 101 60 125)(21 102 61 126)(22 103 62 127)(23 104 63 128)(24 105 64 129)(25 106 65 130)(26 107 66 131)(27 108 67 132)(28 109 68 133)(29 110 69 134)(30 111 70 135)(31 112 71 136)(32 113 72 137)(33 114 73 138)(34 115 74 139)(35 116 75 140)(36 117 76 141)(37 118 77 142)(38 119 78 143)(39 120 79 144)(40 81 80 145)
(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 80 41 40)(2 39 42 79)(3 78 43 38)(4 37 44 77)(5 76 45 36)(6 35 46 75)(7 74 47 34)(8 33 48 73)(9 72 49 32)(10 31 50 71)(11 70 51 30)(12 29 52 69)(13 68 53 28)(14 27 54 67)(15 66 55 26)(16 25 56 65)(17 64 57 24)(18 23 58 63)(19 62 59 22)(20 21 60 61)(81 146 145 82)(83 144 147 120)(84 119 148 143)(85 142 149 118)(86 117 150 141)(87 140 151 116)(88 115 152 139)(89 138 153 114)(90 113 154 137)(91 136 155 112)(92 111 156 135)(93 134 157 110)(94 109 158 133)(95 132 159 108)(96 107 160 131)(97 130 121 106)(98 105 122 129)(99 128 123 104)(100 103 124 127)(101 126 125 102)

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

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

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

86 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F20A···20X40A···40AF
order1222224···444558···810···1020···2040···40
size111140402···24040222···22···22···22···2

86 irreducible representations

dim1111122222222222
type++++++++++++
imageC1C2C2C2C2D4D5D8SD16C4○D4D10D10D20C40⋊C2D40C4○D20
kernelC4.5D40D205C4C4×C40C202Q8C204D4C2×C20C4×C8C20C20C20C42C2×C8C2×C4C4C4C4
# reps1411122444248161616

Matrix representation of C4.5D40 in GL4(𝔽41) generated by

302800
221100
00400
00040
,
221300
19000
00011
002617
,
193200
222200
00011
00150
G:=sub<GL(4,GF(41))| [30,22,0,0,28,11,0,0,0,0,40,0,0,0,0,40],[22,19,0,0,13,0,0,0,0,0,0,26,0,0,11,17],[19,22,0,0,32,22,0,0,0,0,0,15,0,0,11,0] >;

C4.5D40 in GAP, Magma, Sage, TeX

C_4._5D_{40}
% in TeX

G:=Group("C4.5D40");
// GroupNames label

G:=SmallGroup(320,321);
// by ID

G=gap.SmallGroup(320,321);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,142,1123,136,12550]);
// Polycyclic

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

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