Copied to
clipboard

## G = C8×F5order 160 = 25·5

### Direct product of C8 and F5

Aliases: C8×F5, C403C4, C10.1C42, C5⋊C83C4, C51(C4×C8), D5.(C2×C8), C52C87C4, C2.1(C4×F5), D5⋊C8.3C2, (C8×D5).9C2, (C2×F5).2C4, (C4×F5).3C2, C4.16(C2×F5), C20.15(C2×C4), D10.5(C2×C4), Dic5.7(C2×C4), (C4×D5).31C22, SmallGroup(160,66)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C8×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — C8×F5
 Lower central C5 — C8×F5
 Upper central C1 — C8

Generators and relations for C8×F5
G = < a,b,c | a8=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C8×F5
On 40 points
Generators in S40
(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)
(1 26 22 9 35)(2 27 23 10 36)(3 28 24 11 37)(4 29 17 12 38)(5 30 18 13 39)(6 31 19 14 40)(7 32 20 15 33)(8 25 21 16 34)
(1 7 5 3)(2 8 6 4)(9 32 18 37)(10 25 19 38)(11 26 20 39)(12 27 21 40)(13 28 22 33)(14 29 23 34)(15 30 24 35)(16 31 17 36)

G:=sub<Sym(40)| (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), (1,26,22,9,35)(2,27,23,10,36)(3,28,24,11,37)(4,29,17,12,38)(5,30,18,13,39)(6,31,19,14,40)(7,32,20,15,33)(8,25,21,16,34), (1,7,5,3)(2,8,6,4)(9,32,18,37)(10,25,19,38)(11,26,20,39)(12,27,21,40)(13,28,22,33)(14,29,23,34)(15,30,24,35)(16,31,17,36)>;

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), (1,26,22,9,35)(2,27,23,10,36)(3,28,24,11,37)(4,29,17,12,38)(5,30,18,13,39)(6,31,19,14,40)(7,32,20,15,33)(8,25,21,16,34), (1,7,5,3)(2,8,6,4)(9,32,18,37)(10,25,19,38)(11,26,20,39)(12,27,21,40)(13,28,22,33)(14,29,23,34)(15,30,24,35)(16,31,17,36) );

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)], [(1,26,22,9,35),(2,27,23,10,36),(3,28,24,11,37),(4,29,17,12,38),(5,30,18,13,39),(6,31,19,14,40),(7,32,20,15,33),(8,25,21,16,34)], [(1,7,5,3),(2,8,6,4),(9,32,18,37),(10,25,19,38),(11,26,20,39),(12,27,21,40),(13,28,22,33),(14,29,23,34),(15,30,24,35),(16,31,17,36)]])

C8×F5 is a maximal subgroup of   C167F5  C20.12C42  M4(2)⋊5F5  D85F5  SD163F5  Q165F5  C30.C42
C8×F5 is a maximal quotient of   C167F5  C20.31M4(2)  D10.3M4(2)  C30.C42

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 5 8A 8B 8C 8D 8E ··· 8P 10 20A 20B 40A 40B 40C 40D order 1 2 2 2 4 4 4 ··· 4 5 8 8 8 8 8 ··· 8 10 20 20 40 40 40 40 size 1 1 5 5 1 1 5 ··· 5 4 1 1 1 1 5 ··· 5 4 4 4 4 4 4 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 F5 C2×F5 C4×F5 C8×F5 kernel C8×F5 C8×D5 D5⋊C8 C4×F5 C5⋊2C8 C40 C5⋊C8 C2×F5 F5 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 2 4

Matrix representation of C8×F5 in GL5(𝔽41)

 27 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 40 40 40 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 40 40 40 40

G:=sub<GL(5,GF(41))| [27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,40,0,0,0],[1,0,0,0,0,0,1,0,0,40,0,0,0,1,40,0,0,0,0,40,0,0,1,0,40] >;

C8×F5 in GAP, Magma, Sage, TeX

C_8\times F_5
% in TeX

G:=Group("C8xF5");
// GroupNames label

G:=SmallGroup(160,66);
// by ID

G=gap.SmallGroup(160,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,69,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^8=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

׿
×
𝔽