direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×F5, C40⋊3C4, C10.1C42, C5⋊C8⋊3C4, C5⋊1(C4×C8), D5.(C2×C8), C5⋊2C8⋊7C4, 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
| C5 — C8×F5 |
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
(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
C16⋊7F5 C20.12C42 M4(2)⋊5F5 D8⋊5F5 SD16⋊3F5 Q16⋊5F5 C30.C42
C8×F5 is a maximal quotient of C16⋊7F5 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