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G = M4(2)×F5order 320 = 26·5

Direct product of M4(2) and F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)×F5, C20.6C42, D10.2C42, Dic5.2C42, C86(C2×F5), C406(C2×C4), (C8×F5)⋊7C2, C4.F53C4, C8⋊D52C4, C8⋊F56C2, (C4×F5).3C4, C52(C4×M4(2)), C4.11(C4×F5), C4.Dic54C4, C22.F56C4, (C5×M4(2))⋊5C4, (C2×C10).6C42, (C22×F5).4C4, C22.11(C4×F5), C4.52(C22×F5), C10.14(C2×C42), C20.92(C22×C4), D5⋊C8.19C22, D5⋊M4(2).3C2, (C4×D5).88C23, (C8×D5).36C22, D5.2(C2×M4(2)), (C4×F5).18C22, D10.34(C22×C4), (D5×M4(2)).10C2, Dic5.33(C22×C4), C5⋊C82(C2×C4), (C2×C4×F5).4C2, C2.15(C2×C4×F5), C52C816(C2×C4), (C2×F5).7(C2×C4), (C2×C4).76(C2×F5), (C2×C20).45(C2×C4), (C4×D5).46(C2×C4), (C2×C4×D5).192C22, (C2×Dic5).66(C2×C4), (C22×D5).52(C2×C4), SmallGroup(320,1064)

Series: Derived Chief Lower central Upper central

C1C10 — M4(2)×F5
C1C5C10D10C4×D5C4×F5C2×C4×F5 — M4(2)×F5
C5C10 — M4(2)×F5
C1C4M4(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 [×4], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×13], C23, D5 [×2], D5, C10, C10, C42 [×4], C2×C8 [×4], M4(2), M4(2) [×7], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], F5 [×2], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C4×M4(2), C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), D5⋊C8 [×2], C4.F5 [×2], C4×F5 [×2], C4×F5 [×2], C22.F5 [×2], C2×C4×D5, C22×F5 [×2], C8×F5 [×2], C8⋊F5 [×2], D5×M4(2), D5⋊M4(2), C2×C4×F5, M4(2)×F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C2×C42, C2×M4(2) [×2], C2×F5 [×3], C4×M4(2), C4×F5 [×2], 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 2A2B2C2D2E4A4B4C4D···4M4N···4R 5 8A8B8C8D8E···8P10A10B20A20B20C40A40B40C40D
order1222224444···44···4588888···8101020202040404040
size11255101125···510···104222210···10484488888

50 irreducible representations

dim11111111111112444448
type+++++++++
imageC1C2C2C2C2C2C4C4C4C4C4C4C4M4(2)F5C2×F5C2×F5C4×F5C4×F5M4(2)×F5
kernelM4(2)×F5C8×F5C8⋊F5D5×M4(2)D5⋊M4(2)C2×C4×F5C8⋊D5C4.Dic5C5×M4(2)C4.F5C4×F5C22.F5C22×F5F5M4(2)C8C2×C4C4C22C1
# reps12211142244448121222

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

40390000
3710000
001000
000100
000010
000001
,
100000
40400000
0040000
0004000
0000400
0000040
,
100000
010000
0040404040
001000
000100
000010
,
3200000
0320000
001000
000001
000100
0040404040

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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