Copied to
clipboard

G = M4(2)⋊F5  order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊2F5, C20.3(C4⋊C4), (C4×D5).5Q8, (C22×F5).C4, C4.18(C4⋊F5), C4.Dic5⋊1C4, C22.F5⋊3C4, D10.6(C4⋊C4), C22.8(C4×F5), (C4×D5).110D4, (C5×M4(2))⋊2C4, D5.(C4.D4), (C2×C10).3C42, D5.(C4.10D4), D5⋊M4(2).1C2, (D5×M4(2)).3C2, C4.28(C22⋊F5), C20.28(C22⋊C4), C5⋊1(C22.C42), Dic5.25(C4⋊C4), D10.32(C22⋊C4), Dic5.5(C22⋊C4), C2.14(D10.3Q8), C10.13(C2.C42), (C2×C4⋊F5).2C2, (C2×C4).12(C2×F5), (C2×C20).35(C2×C4), (C2×C4×D5).186C22, (C2×Dic5).43(C2×C4), (C22×D5).36(C2×C4), SmallGroup(320,237)

Series: Derived ►Chief ►Lower central ►Upper central

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

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

Subgroups: 418 in 98 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C4⋊C4, C2×M4(2), C5⋊2C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22.C42, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8, C4.F5, C4⋊F5, C22.F5, C2×C4×D5, C22×F5, D5×M4(2), D5⋊M4(2), C2×C4⋊F5, M4(2)⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C4.D4, C4.10D4, C2×F5, C22.C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, 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)

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H 5 8A8B8C···8H10A10B20A20B20C40A40B40C40D
order122222444444445888···8101020202040404040
size11255102210102020202044420···20484488888

32 irreducible representations

dim111111112244444448
type+++++-++-++
imageC1C2C2C2C4C4C4C4D4Q8F5C4.D4C4.10D4C2×F5C4⋊F5C22⋊F5C4×F5M4(2)⋊F5
kernelM4(2)⋊F5D5×M4(2)D5⋊M4(2)C2×C4⋊F5C4.Dic5C5×M4(2)C22.F5C22×F5C4×D5C4×D5M4(2)D5D5C2×C4C4C4C22C1
# reps111122443111112222

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

90000000
09000000
00900000
00090000
0000364390
0000436039
000001537
000000375
,
10000000
01000000
00100000
00010000
00001000
00000100
0000364400
0000436040
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
10000000
00010000
01000000
404040400000
00001000
000004000
000003710
000040040

G:=sub<GL(8,GF(41))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,36,4,0,0,0,0,0,0,4,36,1,0,0,0,0,0,39,0,5,37,0,0,0,0,0,39,37,5],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,4,0,0,0,0,0,1,4,36,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[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],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,40,37,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;
 

M4(2)⋊F5 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes F_5
 
% in TeX
 
G:=Group("M4(2):F5");
 
// GroupNames label
 
G:=SmallGroup(320,237);
 
// by ID
 
G=gap.SmallGroup(320,237);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,136,851,6278,3156]);
 
// 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*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁