M4(2)xF5 - GroupNames

G = M₄(2)×F₅ order 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, C₈⋊₆(C₂×F₅), C₄₀⋊₆(C₂×C₄), (C₈×F₅)⋊₇C₂, C₄.F₅⋊₃C₄, C₈⋊D₅⋊₂C₄, C₈⋊F₅⋊₆C₂, (C₄×F₅).₃C₄, C₅⋊₂(C₄×M₄(2)), C₄.₁₁(C₄×F₅), C₄.Dic₅⋊₄C₄, C₂².F₅⋊₆C₄, (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⋊C₈⋊₂(C₂×C₄), (C₂×C₄×F₅).₄C₂, C₂.₁₅(C₂×C₄×F₅), C₅⋊₂C₈⋊₁₆(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

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), C₅⋊₂C₈, 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

G = M4(2)×F5 order 320 = 26·5

Direct product of M4(2) and F5

G = M₄(2)×F₅ order 320 = 2⁶·5

Direct product of M₄(2) and F₅