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G = C42.20D10order 320 = 26·5

20th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.20D10, C8⋊C411D5, C202Q83C2, (C2×C20).38D4, (C2×C4).27D20, (C2×C8).160D10, (C4×C20).5C22, C2.9(C8⋊D10), C10.6(C8⋊C22), C4.D20.4C2, C22.99(C2×D20), D205C4.16C2, C20.225(C4○D4), C4.109(C4○D20), C20.44D438C2, (C2×C20).735C23, (C2×C40).314C22, C10.9(C4.4D4), C2.8(C8.D10), (C2×D20).10C22, C10.4(C8.C22), C4⋊Dic5.10C22, C2.14(C4.D20), C51(C42.28C22), (C2×Dic10).10C22, (C5×C8⋊C4)⋊20C2, (C2×C10).118(C2×D4), (C2×C4).679(C22×D5), SmallGroup(320,341)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.20D10
C1C5C10C20C2×C20C2×D20C4.D20 — C42.20D10
C5C10C2×C20 — C42.20D10
C1C22C42C8⋊C4

Generators and relations for C42.20D10
 G = < a,b,c,d | a4=b4=1, c10=a2b-1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 494 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×D4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C40 [×2], Dic10 [×4], D20 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.28C22, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×Dic10, C2×Dic10, C2×D20, C20.44D4 [×2], D205C4 [×2], C5×C8⋊C4, C202Q8, C4.D20, C42.20D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22, C8.C22, D20 [×2], C22×D5, C42.28C22, C2×D20, C4○D20 [×2], C4.D20, C8⋊D10, C8.D10, C42.20D10

Smallest permutation representation of C42.20D10
On 160 points
Generators in S160
(1 109 52 159)(2 90 53 140)(3 111 54 121)(4 92 55 142)(5 113 56 123)(6 94 57 144)(7 115 58 125)(8 96 59 146)(9 117 60 127)(10 98 61 148)(11 119 62 129)(12 100 63 150)(13 81 64 131)(14 102 65 152)(15 83 66 133)(16 104 67 154)(17 85 68 135)(18 106 69 156)(19 87 70 137)(20 108 71 158)(21 89 72 139)(22 110 73 160)(23 91 74 141)(24 112 75 122)(25 93 76 143)(26 114 77 124)(27 95 78 145)(28 116 79 126)(29 97 80 147)(30 118 41 128)(31 99 42 149)(32 120 43 130)(33 101 44 151)(34 82 45 132)(35 103 46 153)(36 84 47 134)(37 105 48 155)(38 86 49 136)(39 107 50 157)(40 88 51 138)
(1 42 21 62)(2 43 22 63)(3 44 23 64)(4 45 24 65)(5 46 25 66)(6 47 26 67)(7 48 27 68)(8 49 28 69)(9 50 29 70)(10 51 30 71)(11 52 31 72)(12 53 32 73)(13 54 33 74)(14 55 34 75)(15 56 35 76)(16 57 36 77)(17 58 37 78)(18 59 38 79)(19 60 39 80)(20 61 40 41)(81 121 101 141)(82 122 102 142)(83 123 103 143)(84 124 104 144)(85 125 105 145)(86 126 106 146)(87 127 107 147)(88 128 108 148)(89 129 109 149)(90 130 110 150)(91 131 111 151)(92 132 112 152)(93 133 113 153)(94 134 114 154)(95 135 115 155)(96 136 116 156)(97 137 117 157)(98 138 118 158)(99 139 119 159)(100 140 120 160)
(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 10 52 61)(2 60 53 9)(3 8 54 59)(4 58 55 7)(5 6 56 57)(11 40 62 51)(12 50 63 39)(13 38 64 49)(14 48 65 37)(15 36 66 47)(16 46 67 35)(17 34 68 45)(18 44 69 33)(19 32 70 43)(20 42 71 31)(21 30 72 41)(22 80 73 29)(23 28 74 79)(24 78 75 27)(25 26 76 77)(81 136 131 86)(82 85 132 135)(83 134 133 84)(87 130 137 120)(88 119 138 129)(89 128 139 118)(90 117 140 127)(91 126 141 116)(92 115 142 125)(93 124 143 114)(94 113 144 123)(95 122 145 112)(96 111 146 121)(97 160 147 110)(98 109 148 159)(99 158 149 108)(100 107 150 157)(101 156 151 106)(102 105 152 155)(103 154 153 104)

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim11111122222224444
type++++++++++++-+-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D20C4○D20C8⋊C22C8.C22C8⋊D10C8.D10
kernelC42.20D10C20.44D4D205C4C5×C8⋊C4C202Q8C4.D20C2×C20C8⋊C4C20C42C2×C8C2×C4C4C10C10C2C2
# reps122111224248161144

Matrix representation of C42.20D10 in GL6(𝔽41)

40210000
3710000
0027131819
0028231540
00222528
008202227
,
4000000
0400000
0021300
00283900
00001113
00001930
,
9160000
36320000
001313018
002892023
0021141928
0037220
,
32250000
090000
002120147
002320135
001538323
002363738

G:=sub<GL(6,GF(41))| [40,37,0,0,0,0,21,1,0,0,0,0,0,0,27,28,22,8,0,0,13,23,2,20,0,0,18,15,5,22,0,0,19,40,28,27],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,11,19,0,0,0,0,13,30],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,13,28,21,3,0,0,13,9,14,7,0,0,0,20,19,22,0,0,18,23,28,0],[32,0,0,0,0,0,25,9,0,0,0,0,0,0,21,23,15,23,0,0,20,20,38,6,0,0,14,1,3,37,0,0,7,35,23,38] >;

C42.20D10 in GAP, Magma, Sage, TeX

C_4^2._{20}D_{10}
% in TeX

G:=Group("C4^2.20D10");
// GroupNames label

G:=SmallGroup(320,341);
// by ID

G=gap.SmallGroup(320,341);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^-1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations

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