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G = C20.65(C4⋊C4)  order 320 = 26·5

12nd non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.65(C4⋊C4), C20.86(C2×Q8), (C2×C20).60Q8, (C22×C8).6D5, C20.433(C2×D4), (C2×C8).291D10, (C2×C20).481D4, C4⋊Dic5.25C4, (C22×C40).9C2, C20.8Q82C2, C23.32(C4×D5), C10.40(C8○D4), (C2×C4).47Dic10, C4.51(C2×Dic10), C23.D5.15C4, (C2×C20).857C23, (C2×C40).351C22, (C22×C4).421D10, C4.26(C10.D4), C55(C42.6C22), C2.17(D20.3C4), (C22×C20).538C22, (C4×Dic5).206C22, C23.21D10.4C2, C22.16(C10.D4), C10.66(C2×C4⋊C4), (C2×C4).113(C4×D5), C4.123(C2×C5⋊D4), (C2×C10).76(C4⋊C4), C22.140(C2×C4×D5), (C2×C20).399(C2×C4), (C2×C4.Dic5).4C2, (C2×C4).251(C5⋊D4), (C2×Dic5).32(C2×C4), C2.15(C2×C10.D4), (C2×C4).799(C22×D5), (C2×C10).228(C22×C4), (C22×C10).160(C2×C4), (C2×C52C8).207C22, SmallGroup(320,729)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.65(C4⋊C4)
C1C5C10C20C2×C20C4×Dic5C23.21D10 — C20.65(C4⋊C4)
C5C2×C10 — C20.65(C4⋊C4)
C1C2×C4C22×C8

Generators and relations for C20.65(C4⋊C4)
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, ac=ca, cbc-1=a10b-1 >

Subgroups: 286 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, 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], M4(2) [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C42.6C22, C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C2×C40 [×2], C22×C20, C20.8Q8 [×4], C2×C4.Dic5, C23.21D10, C22×C40, C20.65(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C8○D4 [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C42.6C22, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, D20.3C4 [×2], C2×C10.D4, C20.65(C4⋊C4)

Smallest permutation representation of C20.65(C4⋊C4)
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)
(1 108 62 152)(2 107 63 151)(3 106 64 150)(4 105 65 149)(5 104 66 148)(6 103 67 147)(7 102 68 146)(8 101 69 145)(9 120 70 144)(10 119 71 143)(11 118 72 142)(12 117 73 141)(13 116 74 160)(14 115 75 159)(15 114 76 158)(16 113 77 157)(17 112 78 156)(18 111 79 155)(19 110 80 154)(20 109 61 153)(21 54 125 90)(22 53 126 89)(23 52 127 88)(24 51 128 87)(25 50 129 86)(26 49 130 85)(27 48 131 84)(28 47 132 83)(29 46 133 82)(30 45 134 81)(31 44 135 100)(32 43 136 99)(33 42 137 98)(34 41 138 97)(35 60 139 96)(36 59 140 95)(37 58 121 94)(38 57 122 93)(39 56 123 92)(40 55 124 91)
(1 50 6 55 11 60 16 45)(2 51 7 56 12 41 17 46)(3 52 8 57 13 42 18 47)(4 53 9 58 14 43 19 48)(5 54 10 59 15 44 20 49)(21 109 36 104 31 119 26 114)(22 110 37 105 32 120 27 115)(23 111 38 106 33 101 28 116)(24 112 39 107 34 102 29 117)(25 113 40 108 35 103 30 118)(61 85 66 90 71 95 76 100)(62 86 67 91 72 96 77 81)(63 87 68 92 73 97 78 82)(64 88 69 93 74 98 79 83)(65 89 70 94 75 99 80 84)(121 149 136 144 131 159 126 154)(122 150 137 145 132 160 127 155)(123 151 138 146 133 141 128 156)(124 152 139 147 134 142 129 157)(125 153 140 148 135 143 130 158)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L10A···10N20A···20P40A···40AF
order1222224444444444558···8888810···1020···2040···40
size11112211112220202020222···2202020202···22···22···2

92 irreducible representations

dim111111122222222222
type++++++-+++-
imageC1C2C2C2C2C4C4D4Q8D5D10D10C8○D4Dic10C4×D5C5⋊D4C4×D5D20.3C4
kernelC20.65(C4⋊C4)C20.8Q8C2×C4.Dic5C23.21D10C22×C40C4⋊Dic5C23.D5C2×C20C2×C20C22×C8C2×C8C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps1411144222428848432

Matrix representation of C20.65(C4⋊C4) in GL4(𝔽41) generated by

20000
03900
002137
0002
,
03200
32000
003234
00299
,
27000
02700
003825
0003
G:=sub<GL(4,GF(41))| [20,0,0,0,0,39,0,0,0,0,21,0,0,0,37,2],[0,32,0,0,32,0,0,0,0,0,32,29,0,0,34,9],[27,0,0,0,0,27,0,0,0,0,38,0,0,0,25,3] >;

C20.65(C4⋊C4) in GAP, Magma, Sage, TeX

C_{20}._{65}(C_4\rtimes C_4)
% in TeX

G:=Group("C20.65(C4:C4)");
// GroupNames label

G:=SmallGroup(320,729);
// by ID

G=gap.SmallGroup(320,729);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,136,12550]);
// 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,c*b*c^-1=a^10*b^-1>;
// generators/relations

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