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G = C40.1C8order 320 = 26·5

1st non-split extension by C40 of C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C40.1C8, C8.1(C5⋊C8), (C2×C8).12F5, C10.8(C4⋊C8), C52(C8.C8), (C2×C40).12C4, C20.42(C2×C8), C52C8.39D4, C4.27(C4⋊F5), C20.27(C4⋊C4), C52C8.13Q8, C2.5(C20⋊C8), C20.C8.3C2, (C4×Dic5).39C4, (C8×Dic5).20C2, (C2×C10).3M4(2), C22.2(C4.F5), C4.8(C2×C5⋊C8), (C2×C4).121(C2×F5), (C2×C20).138(C2×C4), (C2×C52C8).337C22, SmallGroup(320,227)

Series: Derived Chief Lower central Upper central

C1C20 — C40.1C8
C1C5C10C20C52C8C2×C52C8C20.C8 — C40.1C8
C5C10C20 — C40.1C8
C1C4C2×C4C2×C8

Generators and relations for C40.1C8
 G = < a,b | a40=1, b8=a20, bab-1=a23 >

2C2
10C4
10C4
2C10
5C8
5C8
10C2×C4
2Dic5
2Dic5
5C2×C8
5C42
10C16
10C16
2C2×Dic5
5M5(2)
5C4×C8
5M5(2)
2C5⋊C16
2C5⋊C16
5C8.C8

Smallest permutation representation of C40.1C8
On 80 points
Generators in S80
(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)
(1 47 6 42 11 77 16 72 21 67 26 62 31 57 36 52)(2 54 15 65 12 44 25 55 22 74 35 45 32 64 5 75)(3 61 24 48 13 51 34 78 23 41 4 68 33 71 14 58)(7 49 20 60 17 79 30 50 27 69 40 80 37 59 10 70)(8 56 29 43 18 46 39 73 28 76 9 63 38 66 19 53)

G:=sub<Sym(80)| (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), (1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52)(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75)(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58)(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70)(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53)>;

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), (1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52)(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75)(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58)(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70)(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53) );

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)], [(1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52),(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75),(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58),(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70),(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53)]])

44 conjugacy classes

class 1 2A2B4A4B4C4D4E4F4G 5 8A8B8C8D8E8F8G8H8I8J10A10B10C16A···16H20A20B20C20D40A···40H
order12244444445888888888810101016···162020202040···40
size11211210101010422225555101044420···2044444···4

44 irreducible representations

dim1111112222444444
type++++-+-+
imageC1C2C2C4C4C8D4Q8M4(2)C8.C8F5C5⋊C8C2×F5C4⋊F5C4.F5C40.1C8
kernelC40.1C8C8×Dic5C20.C8C4×Dic5C2×C40C40C52C8C52C8C2×C10C5C2×C8C8C2×C4C4C22C1
# reps1122281128121228

Matrix representation of C40.1C8 in GL4(𝔽241) generated by

1677400
167800
000211
003084
,
0010
0001
816700
023300
G:=sub<GL(4,GF(241))| [167,167,0,0,74,8,0,0,0,0,0,30,0,0,211,84],[0,0,8,0,0,0,167,233,1,0,0,0,0,1,0,0] >;

C40.1C8 in GAP, Magma, Sage, TeX

C_{40}._1C_8
% in TeX

G:=Group("C40.1C8");
// GroupNames label

G:=SmallGroup(320,227);
// by ID

G=gap.SmallGroup(320,227);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,102,6278,3156]);
// Polycyclic

G:=Group<a,b|a^40=1,b^8=a^20,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C40.1C8 in TeX

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