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## G = M4(2)×F5order 320 = 26·5

### Direct product of M4(2) and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — M4(2)×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — C2×C4×F5 — M4(2)×F5
 Lower central C5 — C10 — M4(2)×F5
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2)×F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 442 in 142 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, F5, D10, D10, C2×C10, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×M4(2), C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8, C4.F5, C4×F5, C4×F5, C22.F5, C2×C4×D5, C22×F5, C8×F5, C8⋊F5, D5×M4(2), D5⋊M4(2), C2×C4×F5, M4(2)×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, F5, C2×C42, C2×M4(2), C2×F5, C4×M4(2), C4×F5, C22×F5, C2×C4×F5, M4(2)×F5

Smallest permutation representation of M4(2)×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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
(1 35 24 12 26)(2 36 17 13 27)(3 37 18 14 28)(4 38 19 15 29)(5 39 20 16 30)(6 40 21 9 31)(7 33 22 10 32)(8 34 23 11 25)
(1 7 5 3)(2 8 6 4)(9 38 17 25)(10 39 18 26)(11 40 19 27)(12 33 20 28)(13 34 21 29)(14 35 22 30)(15 36 23 31)(16 37 24 32)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 8 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4 M4(2) F5 C2×F5 C2×F5 C4×F5 C4×F5 M4(2)×F5 kernel M4(2)×F5 C8×F5 C8⋊F5 D5×M4(2) D5⋊M4(2) C2×C4×F5 C8⋊D5 C4.Dic5 C5×M4(2) C4.F5 C4×F5 C22.F5 C22×F5 F5 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 4 4 8 1 2 1 2 2 2

Matrix representation of M4(2)×F5 in GL6(𝔽41)

 40 39 0 0 0 0 37 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40

G:=sub<GL(6,GF(41))| [40,37,0,0,0,0,39,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

M4(2)×F5 in GAP, Magma, Sage, TeX

M_4(2)\times F_5
% in TeX

G:=Group("M4(2)xF5");
// GroupNames label

G:=SmallGroup(320,1064);
// by ID

G=gap.SmallGroup(320,1064);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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