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G = C80.2C4order 320 = 26·5

2nd non-split extension by C80 of C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C80.2C4, C16.2F5, D10.4Q16, Dic5.11D8, C4.6(C4⋊F5), C8.23(C2×F5), C52C8.9Q8, C52C16.4C4, C40.23(C2×C4), (C4×D5).77D4, (D5×C16).6C2, C20.13(C4⋊C4), C52(C8.4Q8), C2.7(D5.D8), C10.4(C2.D8), D10.Q8.2C2, (C8×D5).52C22, SmallGroup(320,190)

Series: Derived Chief Lower central Upper central

C1C40 — C80.2C4
C1C5C10C20C4×D5C8×D5D10.Q8 — C80.2C4
C5C10C20C40 — C80.2C4
C1C2C4C8C16

Generators and relations for C80.2C4
 G = < a,b | a80=1, b4=a40, bab-1=a23 >

10C2
5C4
5C22
2D5
5C8
5C2×C4
20C8
20C8
5C2×C8
5C16
10M4(2)
10M4(2)
4C5⋊C8
4C5⋊C8
5C2×C16
5C8.C4
5C8.C4
2C4.F5
2C4.F5
5C8.4Q8

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

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

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

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

38 conjugacy classes

class 1 2A2B4A4B4C 5 8A8B8C8D8E8F8G8H 10 16A16B16C16D16E16F16G16H20A20B40A40B40C40D80A···80H
order12244458888888810161616161616161620204040404080···80
size111025542210104040404042222101010104444444···4

38 irreducible representations

dim111112222244444
type+++-++-++
imageC1C2C2C4C4Q8D4D8Q16C8.4Q8F5C2×F5C4⋊F5D5.D8C80.2C4
kernelC80.2C4D5×C16D10.Q8C52C16C80C52C8C4×D5Dic5D10C5C16C8C4C2C1
# reps112221122811248

Matrix representation of C80.2C4 in GL4(𝔽7) generated by

2616
5614
2602
2024
,
1354
0443
2333
4146
G:=sub<GL(4,GF(7))| [2,5,2,2,6,6,6,0,1,1,0,2,6,4,2,4],[1,0,2,4,3,4,3,1,5,4,3,4,4,3,3,6] >;

C80.2C4 in GAP, Magma, Sage, TeX

C_{80}._2C_4
% in TeX

G:=Group("C80.2C4");
// GroupNames label

G:=SmallGroup(320,190);
// by ID

G=gap.SmallGroup(320,190);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,1571,80,1684,102,6278,3156]);
// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^40,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C80.2C4 in TeX

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