M4(2):3F5 - GroupNames

G = M₄(2)⋊₃F₅ order 320 = 2⁶·5

3^rd semidirect product of M₄(2) and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2)⋊₃F₅, C₂₀.₃C₄², C₄⋊F₅⋊₂C₄, (C₄×F₅)⋊₄C₄, C₄.₈(C₄×F₅), D₅.₂C₄≀C₂, C₄.Dic₅⋊₂C₄, C₅⋊₃(C₄²⋊₆C₄), (C₄×D₅).₁₁₇D₄, (C₅×M₄(2))⋊₃C₄, Dic₅.₆(C₄⋊C₄), (C₂×Dic₅).₁₁Q₈, (C₂²×D₅).₅₈D₄, (D₅×M₄(2)).₄C₂, C₂².₁₀(C₄⋊F₅), C₄.₄₀(C₂²⋊F₅), C₂₀.₃₈(C₂²⋊C₄), D₁₀.₆(C₂²⋊C₄), C₂.₁₅(D₁₀.₃Q₈), C₁₀.₁₄(C₂.C₄²), D₁₀.C₂³.₁C₂, (C₂×C₄×F₅).₂C₂, (C₂×C₄).₆₉(C₂×F₅), (C₂×C₁₀).₃(C₄⋊C₄), (C₂×C₂₀).₃₆(C₂×C₄), (C₄×D₅).₁₄(C₂×C₄), (C₂×C₄×D₅).₁₈₇C₂², SmallGroup(320,238)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — M₄(2)⋊₃F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₂×C₄×D₅ — D₁₀.C₂³ — M₄(2)⋊₃F₅

Lower central

C₅ — C₁₀ — C₂₀ — M₄(2)⋊₃F₅

Upper central

C₁ — C₄ — C₂×C₄ — M₄(2)

Generators and relations for M₄(2)⋊₃F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a⁵, ac=ca, dad^-1=a^-1b, bc=cb, dbd^-1=a⁴b, dcd^-1=c³ >

Subgroups: 466 in 110 conjugacy classes, 36 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, 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, C₂×C₄², C₄²⋊C₂, C₂×M₄(2), C₅⋊₂C₈, C₄₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₄²⋊₆C₄, C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₅×M₄(2), C₄×F₅, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₂×C₄×D₅, C₂²×F₅, D₅×M₄(2), C₂×C₄×F₅, D₁₀.C₂³, M₄(2)⋊₃F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, F₅, C₂.C₄², C₄≀C₂, C₂×F₅, C₄²⋊₆C₄, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₁₀.₃Q₈, M₄(2)⋊₃F₅

Smallest permutation representation of M₄(2)⋊₃F₅
►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 26 22 16 35)(2 27 23 9 36)(3 28 24 10 37)(4 29 17 11 38)(5 30 18 12 39)(6 31 19 13 40)(7 32 20 14 33)(8 25 21 15 34)

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	···	4N	4O	4P	4Q	4R	5	8A	8B	8C	8D	10A	10B	20A	20B	20C	40A	40B	40C	40D
order	1	2	2	2	2	2	4	4	4	4	4	4	···	4	4	4	4	4	5	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	20	20	20	20	4	4	4	20	20	4	8	4	4	8	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	4	8
type	+	+	+	+					+	-	+		+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	Q₈	D₄	C₄≀C₂	F₅	C₂×F₅	C₄×F₅	C₂²⋊F₅	C₄⋊F₅	M₄(2)⋊₃F₅
kernel	M₄(2)⋊₃F₅	D₅×M₄(2)	C₂×C₄×F₅	D₁₀.C₂³	C₄.Dic₅	C₅×M₄(2)	C₄×F₅	C₄⋊F₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	D₅	M₄(2)	C₂×C₄	C₄	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	4	4	2	1	1	8	1	1	2	2	2	2

Matrix representation of M₄(2)⋊₃F₅ ►in GL₆(𝔽₄₁)

28	37	0	0	0	0
40	13	0	0	0	0
0	0	7	27	0	14
0	0	0	34	27	14
0	0	14	27	34	0
0	0	14	0	27	7

1	0	0	0	0	0
14	40	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
0	1	0	0	0	0
0	0	0	0	0	40
0	0	1	0	0	40
0	0	0	1	0	40
0	0	0	0	1	40

13	4	0	0	0	0
19	28	0	0	0	0
0	0	34	0	7	14
0	0	0	14	34	7
0	0	27	7	34	14
0	0	27	14	7	0

G:=sub<GL(6,GF(41))| [28,40,0,0,0,0,37,13,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[1,14,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[13,19,0,0,0,0,4,28,0,0,0,0,0,0,34,0,27,27,0,0,0,14,7,14,0,0,7,34,34,7,0,0,14,7,14,0] >;

M₄(2)⋊₃F₅ in GAP, Magma, Sage, TeX

M_4(2)\rtimes_3F_5

% in TeX

G:=Group("M4(2):3F5");

// GroupNames label

G:=SmallGroup(320,238);

// by ID

G=gap.SmallGroup(320,238);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,1684,851,102,6278,3156]);

// 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^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;

// generators/relations

G = M4(2)⋊3F5 order 320 = 26·5

3rd semidirect product of M4(2) and F5 acting via F5/D5=C2

G = M₄(2)⋊₃F₅ order 320 = 2⁶·5

3^rd semidirect product of M₄(2) and F₅ acting via F₅/D₅=C₂