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G = C20.10C42order 320 = 26·5

3rd non-split extension by C20 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.10C42, (C2×C8).9F5, C4.3(C4×F5), (C2×C40).8C4, C4.F5.2C4, (C4×D5).109D4, D10.22(C4⋊C4), C51(C4.C42), C4.26(C22⋊F5), C22.20(C4⋊F5), C2.3(D10.Q8), C2.3(C40.C4), (C22×D5).17Q8, C10.5(C8.C4), C20.26(C22⋊C4), (C2×Dic5).174D4, Dic5.32(C22⋊C4), C2.12(D10.3Q8), C10.11(C2.C42), (D5×C2×C8).13C2, (C2×C52C8).23C4, (C2×C4.F5).6C2, (C4×D5).63(C2×C4), (C2×C4).127(C2×F5), (C2×C10).14(C4⋊C4), (C2×C20).119(C2×C4), (C2×C4×D5).388C22, SmallGroup(320,234)

Series: Derived Chief Lower central Upper central

C1C20 — C20.10C42
C1C5C10Dic5C4×D5C2×C4×D5C2×C4.F5 — C20.10C42
C5C10C20 — C20.10C42
C1C22C2×C4C2×C8

Generators and relations for C20.10C42
 G = < a,b,c | a20=1, b4=c4=a10, bab-1=a3, ac=ca, cbc-1=a15b >

Subgroups: 322 in 90 conjugacy classes, 38 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10 [×3], C2×C8, C2×C8 [×5], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22×C8, C2×M4(2) [×2], C52C8, C40, C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C4.C42, C8×D5 [×2], C2×C52C8, C2×C40, C4.F5 [×4], C4.F5 [×2], C2×C5⋊C8 [×2], C2×C4×D5, D5×C2×C8, C2×C4.F5 [×2], C20.10C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C8.C4 [×2], C2×F5, C4.C42, C4×F5, C4⋊F5, C22⋊F5, C40.C4, D10.Q8, D10.3Q8, C20.10C42

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F 5 8A8B8C8D8E8F8G8H8I···8P10A10B10C20A20B20C20D40A···40H
order1222224444445888888888···81010102020202040···40
size11111010225555422221010101020···2044444444···4

44 irreducible representations

dim11111122224444444
type+++++-+++
imageC1C2C2C4C4C4D4D4Q8C8.C4F5C2×F5C4×F5C22⋊F5C4⋊F5C40.C4D10.Q8
kernelC20.10C42D5×C2×C8C2×C4.F5C2×C52C8C2×C40C4.F5C4×D5C2×Dic5C22×D5C10C2×C8C2×C4C4C4C22C2C2
# reps11222821181122244

Matrix representation of C20.10C42 in GL6(𝔽41)

3200000
090000
0000040
0010040
0001040
0000140
,
010000
900000
00703421
000211328
0020281321
002021340
,
1400000
0380000
0032000
0003200
0000320
0000032

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,7,0,20,20,0,0,0,21,28,21,0,0,34,13,13,34,0,0,21,28,21,0],[14,0,0,0,0,0,0,38,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;

C20.10C42 in GAP, Magma, Sage, TeX

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

G:=Group("C20.10C4^2");
// GroupNames label

G:=SmallGroup(320,234);
// by ID

G=gap.SmallGroup(320,234);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,1123,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations

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