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G = C20.34M4(2)  order 320 = 26·5

9th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.34M4(2), (C2×Dic5)⋊9C8, (C22×C4).15F5, C23.41(C2×F5), C10.15(C22×C8), (C22×C20).35C4, C22.6(D5⋊C8), (C4×Dic5).31C4, Dic5.19(C2×C8), Dic5⋊C816C2, C54(C42.12C4), C10.26(C2×M4(2)), C4.12(C22.F5), C23.2F5.7C2, Dic5.30(C4○D4), C22.48(C22×F5), (C22×Dic5).32C4, C10.14(C42⋊C2), (C4×Dic5).327C22, (C2×Dic5).345C23, C2.6(D10.C23), (C22×Dic5).273C22, (C4×C5⋊C8)⋊18C2, C2.16(C2×D5⋊C8), (C2×C10).15(C2×C8), (C2×C5⋊C8).38C22, (C2×C4).108(C2×F5), (C2×C4×Dic5).52C2, (C2×C20).107(C2×C4), C2.4(C2×C22.F5), (C22×C10).61(C2×C4), (C2×C10).61(C22×C4), (C2×Dic5).183(C2×C4), SmallGroup(320,1092)

Series: Derived Chief Lower central Upper central

C1C10 — C20.34M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C23.2F5 — C20.34M4(2)
C5C10 — C20.34M4(2)
C1C2×C4C22×C4

Generators and relations for C20.34M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a13, ac=ca, cbc=a10b5 >

Subgroups: 330 in 118 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], Dic5 [×6], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C42.12C4, C4×Dic5 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×C20, C4×C5⋊C8 [×2], Dic5⋊C8 [×2], C23.2F5 [×2], C2×C4×Dic5, C20.34M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5 [×3], C42.12C4, D5⋊C8 [×2], C22.F5 [×2], C22×F5, C2×D5⋊C8, D10.C23, C2×C22.F5, C20.34M4(2)

Smallest permutation representation of C20.34M4(2)
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)
(1 149 115 56 38 95 76 125)(2 146 104 49 39 92 65 138)(3 143 113 42 40 89 74 131)(4 160 102 55 21 86 63 124)(5 157 111 48 22 83 72 137)(6 154 120 41 23 100 61 130)(7 151 109 54 24 97 70 123)(8 148 118 47 25 94 79 136)(9 145 107 60 26 91 68 129)(10 142 116 53 27 88 77 122)(11 159 105 46 28 85 66 135)(12 156 114 59 29 82 75 128)(13 153 103 52 30 99 64 121)(14 150 112 45 31 96 73 134)(15 147 101 58 32 93 62 127)(16 144 110 51 33 90 71 140)(17 141 119 44 34 87 80 133)(18 158 108 57 35 84 69 126)(19 155 117 50 36 81 78 139)(20 152 106 43 37 98 67 132)
(41 140)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 141)(98 142)(99 143)(100 144)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N4O4P4Q4R 5 8A···8P10A···10G20A···20H
order1222224444444···4444458···810···1020···20
size1111221111225···510101010410···104···44···4

56 irreducible representations

dim11111111122444444
type++++++++-
imageC1C2C2C2C2C4C4C4C8C4○D4M4(2)F5C2×F5C2×F5C22.F5D5⋊C8D10.C23
kernelC20.34M4(2)C4×C5⋊C8Dic5⋊C8C23.2F5C2×C4×Dic5C4×Dic5C22×Dic5C22×C20C2×Dic5Dic5C20C22×C4C2×C4C23C4C22C2
# reps122214221644121444

Matrix representation of C20.34M4(2) in GL6(𝔽41)

3200000
0320000
0074000
0084000
00327035
003314734
,
010000
100000
00321390
002720039
0034293820
002201421
,
100000
0400000
001000
000100
00321400
002720040

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,7,8,32,33,0,0,40,40,7,14,0,0,0,0,0,7,0,0,0,0,35,34],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,27,34,2,0,0,21,20,29,20,0,0,39,0,38,14,0,0,0,39,20,21],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,3,27,0,0,0,1,21,20,0,0,0,0,40,0,0,0,0,0,0,40] >;

C20.34M4(2) in GAP, Magma, Sage, TeX

C_{20}._{34}M_4(2)
% in TeX

G:=Group("C20.34M4(2)");
// GroupNames label

G:=SmallGroup(320,1092);
// by ID

G=gap.SmallGroup(320,1092);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,268,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^13,a*c=c*a,c*b*c=a^10*b^5>;
// generators/relations

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