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G = C20.42C42order 320 = 26·5

5th non-split extension by C20 of C42 acting via C42/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.42C42, (C2×C40)⋊24C4, (C2×C8)⋊9Dic5, C408C432C2, C40.121(C2×C4), (C8×Dic5)⋊25C2, (C2×C8).328D10, C4.Dic517C4, C4⋊Dic5.36C4, C4.12(C4×Dic5), C8.27(C2×Dic5), (C22×C8).16D5, C23.31(C4×D5), C10.39(C8○D4), C58(C82M4(2)), C10.44(C2×C42), (C2×C10).48C42, (C22×C40).25C2, C23.D5.21C4, (C2×C40).415C22, C20.198(C22×C4), (C2×C20).856C23, (C22×C4).420D10, C22.12(C4×Dic5), C4.33(C22×Dic5), C2.4(D20.3C4), (C22×C20).537C22, (C4×Dic5).312C22, C23.21D10.26C2, C4.113(C2×C4×D5), C2.12(C2×C4×Dic5), C22.59(C2×C4×D5), C52C8.31(C2×C4), (C2×C4).112(C4×D5), (C2×C20).407(C2×C4), (C2×C4).80(C2×Dic5), (C2×C4).798(C22×D5), (C2×C4.Dic5).31C2, (C2×C10).227(C22×C4), (C22×C10).159(C2×C4), (C2×C52C8).327C22, (C2×Dic5).110(C2×C4), SmallGroup(320,728)

Series: Derived Chief Lower central Upper central

C1C10 — C20.42C42
C1C5C10C2×C10C2×C20C4×Dic5C23.21D10 — C20.42C42
C5C10 — C20.42C42
C1C2×C8C22×C8

Generators and relations for C20.42C42
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, ac=ca, bc=cb >

Subgroups: 286 in 130 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×4], C40 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C82M4(2), C2×C52C8 [×2], C4.Dic5 [×4], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C2×C40 [×4], C22×C20, C8×Dic5 [×2], C408C4 [×2], C2×C4.Dic5, C23.21D10, C22×C40, C20.42C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C42, C8○D4 [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C82M4(2), C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, D20.3C4 [×2], C2×C4×Dic5, C20.42C42

Smallest permutation representation of C20.42C42
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)
(1 117 36 100)(2 116 37 99)(3 115 38 98)(4 114 39 97)(5 113 40 96)(6 112 21 95)(7 111 22 94)(8 110 23 93)(9 109 24 92)(10 108 25 91)(11 107 26 90)(12 106 27 89)(13 105 28 88)(14 104 29 87)(15 103 30 86)(16 102 31 85)(17 101 32 84)(18 120 33 83)(19 119 34 82)(20 118 35 81)(41 159 71 131)(42 158 72 130)(43 157 73 129)(44 156 74 128)(45 155 75 127)(46 154 76 126)(47 153 77 125)(48 152 78 124)(49 151 79 123)(50 150 80 122)(51 149 61 121)(52 148 62 140)(53 147 63 139)(54 146 64 138)(55 145 65 137)(56 144 66 136)(57 143 67 135)(58 142 68 134)(59 141 69 133)(60 160 70 132)
(1 56 6 41 11 46 16 51)(2 57 7 42 12 47 17 52)(3 58 8 43 13 48 18 53)(4 59 9 44 14 49 19 54)(5 60 10 45 15 50 20 55)(21 71 26 76 31 61 36 66)(22 72 27 77 32 62 37 67)(23 73 28 78 33 63 38 68)(24 74 29 79 34 64 39 69)(25 75 30 80 35 65 40 70)(81 137 96 132 91 127 86 122)(82 138 97 133 92 128 87 123)(83 139 98 134 93 129 88 124)(84 140 99 135 94 130 89 125)(85 121 100 136 95 131 90 126)(101 148 116 143 111 158 106 153)(102 149 117 144 112 159 107 154)(103 150 118 145 113 160 108 155)(104 151 119 146 114 141 109 156)(105 152 120 147 115 142 110 157)

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N5A5B8A···8H8I8J8K8L8M···8T10A···10N20A···20P40A···40AF
order1222224444444···4558···888888···810···1020···2040···40
size11112211112210···10221···1222210···102···22···22···2

104 irreducible representations

dim111111111122222222
type+++++++-++
imageC1C2C2C2C2C2C4C4C4C4D5Dic5D10D10C8○D4C4×D5C4×D5D20.3C4
kernelC20.42C42C8×Dic5C408C4C2×C4.Dic5C23.21D10C22×C40C4.Dic5C4⋊Dic5C23.D5C2×C40C22×C8C2×C8C2×C8C22×C4C10C2×C4C23C2
# reps12211184482842812432

Matrix representation of C20.42C42 in GL4(𝔽41) generated by

4000
03100
00330
0005
,
0100
40000
0001
00400
,
9000
0900
00140
00014
G:=sub<GL(4,GF(41))| [4,0,0,0,0,31,0,0,0,0,33,0,0,0,0,5],[0,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,14,0,0,0,0,14] >;

C20.42C42 in GAP, Magma, Sage, TeX

C_{20}._{42}C_4^2
% in TeX

G:=Group("C20.42C4^2");
// GroupNames label

G:=SmallGroup(320,728);
// by ID

G=gap.SmallGroup(320,728);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations

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