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G = M4(2)xF5order 320 = 26·5

Direct product of M4(2) and F5

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

Aliases: M4(2)xF5, C20.6C42, D10.2C42, Dic5.2C42, C8:6(C2xF5), C40:6(C2xC4), (C8xF5):7C2, C4.F5:3C4, C8:D5:2C4, C8:F5:6C2, (C4xF5).3C4, C5:2(C4xM4(2)), C4.11(C4xF5), C4.Dic5:4C4, C22.F5:6C4, (C5xM4(2)):5C4, (C2xC10).6C42, (C22xF5).4C4, C22.11(C4xF5), C4.52(C22xF5), C10.14(C2xC42), C20.92(C22xC4), D5:C8.19C22, D5:M4(2).3C2, (C4xD5).88C23, (C8xD5).36C22, D5.2(C2xM4(2)), (C4xF5).18C22, D10.34(C22xC4), (D5xM4(2)).10C2, Dic5.33(C22xC4), C5:C8:2(C2xC4), (C2xC4xF5).4C2, C2.15(C2xC4xF5), C5:2C8:16(C2xC4), (C2xF5).7(C2xC4), (C2xC4).76(C2xF5), (C2xC20).45(C2xC4), (C4xD5).46(C2xC4), (C2xC4xD5).192C22, (C2xDic5).66(C2xC4), (C22xD5).52(C2xC4), SmallGroup(320,1064)

Series: Derived Chief Lower central Upper central

C1C10 — M4(2)xF5
C1C5C10D10C4xD5C4xF5C2xC4xF5 — M4(2)xF5
C5C10 — M4(2)xF5
C1C4M4(2)

Generators and relations for M4(2)xF5
 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, C2xC4, C2xC4, C23, D5, D5, C10, C10, C42, C2xC8, M4(2), M4(2), C22xC4, Dic5, C20, F5, F5, D10, D10, C2xC10, C4xC8, C8:C4, C2xC42, C2xM4(2), C5:2C8, C40, C5:C8, C4xD5, C2xDic5, C2xC20, C2xF5, C2xF5, C22xD5, C4xM4(2), C8xD5, C8:D5, C4.Dic5, C5xM4(2), D5:C8, C4.F5, C4xF5, C4xF5, C22.F5, C2xC4xD5, C22xF5, C8xF5, C8:F5, D5xM4(2), D5:M4(2), C2xC4xF5, M4(2)xF5
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, M4(2), C22xC4, F5, C2xC42, C2xM4(2), C2xF5, C4xM4(2), C4xF5, C22xF5, C2xC4xF5, M4(2)xF5

Smallest permutation representation of M4(2)xF5
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)F5C2xF5C2xF5C4xF5C4xF5M4(2)xF5
kernelM4(2)xF5C8xF5C8:F5D5xM4(2)D5:M4(2)C2xC4xF5C8:D5C4.Dic5C5xM4(2)C4.F5C4xF5C22.F5C22xF5F5M4(2)C8C2xC4C4C22C1
# reps12211142244448121222

Matrix representation of M4(2)xF5 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)xF5 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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