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G = C80.9C4order 320 = 26·5

4th non-split extension by C80 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C80.9C4, C54M6(2), C20.4C16, C40.12C8, C16.22D10, C16.2Dic5, C80.27C22, C52C325C2, C4.(C52C16), (C2×C16).7D5, C8.3(C52C8), (C2×C80).13C2, (C2×C20).18C8, C20.77(C2×C8), (C2×C10).5C16, (C2×C40).49C4, C10.19(C2×C16), C40.117(C2×C4), C22.(C52C16), C8.22(C2×Dic5), (C2×C8).14Dic5, C2.4(C2×C52C16), C4.15(C2×C52C8), (C2×C4).5(C52C8), SmallGroup(320,57)

Series: Derived Chief Lower central Upper central

C1C10 — C80.9C4
C1C5C10C20C40C80C52C32 — C80.9C4
C5C10 — C80.9C4
C1C16C2×C16

Generators and relations for C80.9C4
 G = < a,b | a80=1, b4=a50, bab-1=a9 >

2C2
2C10
5C32
5C32
5M6(2)

Smallest permutation representation of C80.9C4
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)
(1 146 66 91 51 116 36 141 21 86 6 111 71 136 56 81 41 106 26 131 11 156 76 101 61 126 46 151 31 96 16 121)(2 155 67 100 52 125 37 150 22 95 7 120 72 145 57 90 42 115 27 140 12 85 77 110 62 135 47 160 32 105 17 130)(3 84 68 109 53 134 38 159 23 104 8 129 73 154 58 99 43 124 28 149 13 94 78 119 63 144 48 89 33 114 18 139)(4 93 69 118 54 143 39 88 24 113 9 138 74 83 59 108 44 133 29 158 14 103 79 128 64 153 49 98 34 123 19 148)(5 102 70 127 55 152 40 97 25 122 10 147 75 92 60 117 45 142 30 87 15 112 80 137 65 82 50 107 35 132 20 157)

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

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

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

104 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F10A···10F16A···16H16I16J16K16L20A···20H32A···32P40A···40P80A···80AF
order1224445588888810···1016···161616161620···2032···3240···4080···80
size112112221111222···21···122222···210···102···22···2

104 irreducible representations

dim1111111112222222222
type++++-+-
imageC1C2C2C4C4C8C8C16C16D5Dic5D10Dic5C52C8C52C8M6(2)C52C16C52C16C80.9C4
kernelC80.9C4C52C32C2×C80C80C2×C40C40C2×C20C20C2×C10C2×C16C16C16C2×C8C8C2×C4C5C4C22C1
# reps12122448822224488832

Matrix representation of C80.9C4 in GL2(𝔽641) generated by

433278
0196
,
2179
441639
G:=sub<GL(2,GF(641))| [433,0,278,196],[2,441,179,639] >;

C80.9C4 in GAP, Magma, Sage, TeX

C_{80}._9C_4
% in TeX

G:=Group("C80.9C4");
// GroupNames label

G:=SmallGroup(320,57);
// by ID

G=gap.SmallGroup(320,57);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,58,80,102,12550]);
// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^50,b*a*b^-1=a^9>;
// generators/relations

Export

Subgroup lattice of C80.9C4 in TeX

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