Copied to
clipboard

G = C80.6C4order 320 = 26·5

1st non-split extension by C80 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C80.6C4, C4.18D40, C20.36D8, C40.14Q8, C16.1Dic5, C8.13Dic10, C22.1Dic20, (C2×C80).7C2, (C2×C16).5D5, (C2×C4).73D20, C20.56(C4⋊C4), (C2×C10).7Q16, C54(C8.4Q8), C40.112(C2×C4), (C2×C20).393D4, (C2×C8).311D10, C8.16(C2×Dic5), C2.5(C405C4), C40.6C4.1C2, C4.10(C4⋊Dic5), C10.16(C2.D8), (C2×C40).383C22, SmallGroup(320,64)

Series: Derived Chief Lower central Upper central

C1C40 — C80.6C4
C1C5C10C20C2×C20C2×C40C40.6C4 — C80.6C4
C5C10C20C40 — C80.6C4
C1C4C2×C4C2×C8C2×C16

Generators and relations for C80.6C4
 G = < a,b | a80=1, b4=a40, bab-1=a39 >

2C2
2C10
20C8
20C8
10M4(2)
10M4(2)
4C52C8
4C52C8
5C8.C4
5C8.C4
2C4.Dic5
2C4.Dic5
5C8.4Q8

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

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

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

86 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F8G8H10A···10F16A···16H20A···20H40A···40P80A···80AF
order122444558888888810···1016···1620···2040···4080···80
size112112222222404040402···22···22···22···22···2

86 irreducible representations

dim11112222222222222
type+++-+++--+-++-
imageC1C2C2C4Q8D4D5D8Q16Dic5D10Dic10D20C8.4Q8D40Dic20C80.6C4
kernelC80.6C4C40.6C4C2×C80C80C40C2×C20C2×C16C20C2×C10C16C2×C8C8C2×C4C5C4C22C1
# reps121411222424488832

Matrix representation of C80.6C4 in GL2(𝔽241) generated by

570
9993
,
7728
109164
G:=sub<GL(2,GF(241))| [57,99,0,93],[77,109,28,164] >;

C80.6C4 in GAP, Magma, Sage, TeX

C_{80}._6C_4
% in TeX

G:=Group("C80.6C4");
// GroupNames label

G:=SmallGroup(320,64);
// by ID

G=gap.SmallGroup(320,64);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,675,192,1684,102,12550]);
// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^40,b*a*b^-1=a^39>;
// generators/relations

Export

Subgroup lattice of C80.6C4 in TeX

׿
×
𝔽