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G = M4(2)⋊1F5  order 320 = 26·5

1st semidirect product of M4(2) and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊1F5, C8⋊1(C2×F5), C40⋊5(C2×C4), C8⋊D5⋊1C4, C40⋊C4⋊5C2, D5.D8⋊4C2, C20.6(C4⋊C4), (C4×D5).9Q8, (C4×D5).30D4, C4.21(C4⋊F5), C4.Dic5⋊5C4, D10.32(C2×D4), C5⋊(M4(2)⋊C4), (C5×M4(2))⋊1C4, C4⋊F5.18C22, D10.10(C4⋊C4), C4.41(C22×F5), C20.81(C22×C4), D5.1(C8⋊C22), (C2×Dic5).14Q8, Dic5.16(C2×Q8), (C4×D5).81C23, (C22×D5).63D4, (C8×D5).29C22, (D5×M4(2)).1C2, C22.13(C4⋊F5), Dic5.10(C4⋊C4), D5.1(C8.C22), D10.C23.3C2, (C2×C4⋊F5).4C2, C2.20(C2×C4⋊F5), C5⋊2C8⋊15(C2×C4), C10.17(C2×C4⋊C4), (C2×C4).31(C2×F5), (C2×C10).6(C4⋊C4), (C2×C20).46(C2×C4), (C4×D5).37(C2×C4), (C2×C4×D5).193C22, SmallGroup(320,1065)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — M4(2)⋊1F5
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — M4(2)⋊1F5
Lower central
C5 — C10 — C20 — M4(2)⋊1F5
Upper central
C1 — C2 — C2×C4 — M4(2)

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

Subgroups: 490 in 118 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, M4(2)⋊C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C4×F5, C4⋊F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C22×F5, C40⋊C4, D5.D8, D5×M4(2), C2×C4⋊F5, D10.C23, M4(2)⋊1F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8⋊C22, C8.C22, C2×F5, M4(2)⋊C4, C4⋊F5, C22×F5, C2×C4⋊F5, M4(2)⋊1F5

Smallest permutation representation of M4(2)⋊1F5
►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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4L 5 8A8B8C8D10A10B20A20B20C40A40B40C40D
order12222244444···458888101020202040404040
size112551022101020···204442020484488888

32 irreducible representations

dim111111111222244444448
type+++++++--+++-++
imageC1C2C2C2C2C2C4C4C4D4Q8Q8D4F5C8⋊C22C8.C22C2×F5C2×F5C4⋊F5C4⋊F5M4(2)⋊1F5
kernelM4(2)⋊1F5C40⋊C4D5.D8D5×M4(2)C2×C4⋊F5D10.C23C8⋊D5C4.Dic5C5×M4(2)C4×D5C4×D5C2×Dic5C22×D5M4(2)D5D5C8C2×C4C4C22C1
# reps122111422111111121222

Matrix representation of M4(2)⋊1F5 ►in GL8(𝔽41)

7014140000
27342700000
02734270000
1414070000
0000534360
0000170036
0000229367
000091240
,
400000000
040000000
004000000
000400000
000040000
000004000
0000391110
000026001
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
10000000
00010000
01000000
404040400000
00001000
0000404000
0000204039
00000001

G:=sub<GL(8,GF(41))| [7,27,0,14,0,0,0,0,0,34,27,14,0,0,0,0,14,27,34,0,0,0,0,0,14,0,27,7,0,0,0,0,0,0,0,0,5,17,22,9,0,0,0,0,34,0,9,1,0,0,0,0,36,0,36,24,0,0,0,0,0,36,7,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,39,26,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,40,2,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,39,1] >;
 

M4(2)⋊1F5 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_1F_5
 
% in TeX
 
G:=Group("M4(2):1F5");
 
// GroupNames label
 
G:=SmallGroup(320,1065);
 
// by ID
 
G=gap.SmallGroup(320,1065);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,1684,102,6278,1595]);
 
// 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,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

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