C5:3C2wrC4 - GroupNames

G = C₅⋊₃C₂≀C₄ order 320 = 2⁶·5

The semidirect product of C₅ and C₂≀C₄ acting via C₂≀C₄/C₂³⋊C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅⋊₃C₂≀C₄, C₂³⋊C₄⋊₁D₅, (C₂×C₄).₁D₂₀, (C₂×C₂₀).₁D₄, (C₂×D₄).₁D₁₀, (C₂³×D₅)⋊₁C₄, C₂³.D₅⋊₁C₄, C₂³.₁(C₄×D₅), C₂₀.D₄⋊₁C₂, (C₂²×C₁₀).₈D₄, C₂³⋊D₁₀.₁C₂, (D₄×C₁₀).₁C₂², C₂³.₁(C₅⋊D₄), C₁₀.₂₈(C₂³⋊C₄), C₂².₈(D₁₀⋊C₄), C₂.₈(C₂³.₁D₁₀), (C₅×C₂³⋊C₄)⋊₁C₂, (C₂²×C₁₀).₁(C₂×C₄), (C₂×C₁₀).₆₅(C₂²⋊C₄), SmallGroup(320,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — C₅⋊₃C₂≀C₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₂³⋊D₁₀ — C₅⋊₃C₂≀C₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₅⋊₃C₂≀C₄

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂³⋊C₄

Generators and relations for C₅⋊₃C₂≀C₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcde, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, ef=fe >

Subgroups: 574 in 94 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₅⋊₂C₈, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₂≀C₄, C₄.Dic₅, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂×C₅⋊D₄, D₄×C₁₀, C₂³×D₅, C₂₀.D₄, C₅×C₂³⋊C₄, C₂³⋊D₁₀, C₅⋊₃C₂≀C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₂≀C₄, D₁₀⋊C₄, C₂³.₁D₁₀, C₅⋊₃C₂≀C₄

Smallest permutation representation of C₅⋊₃C₂≀C₄
►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 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 32)(22 31)(23 35)(24 34)(25 33)(26 37)(27 36)(28 40)(29 39)(30 38)

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

(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	5A	5B	8A	8B	10A	10B	10C	···	10H	10I	10J	20A	···	20J
order	1	2	2	2	2	2	2	4	4	4	4	5	5	8	8	10	10	10	···	10	10	10	20	···	20
size	1	1	2	4	4	20	20	4	8	8	40	2	2	40	40	2	2	4	···	4	8	8	8	···	8

35 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+	+	+			+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	D₁₀	D₂₀	C₄×D₅	C₅⋊D₄	C₂³⋊C₄	C₂≀C₄	C₂³.₁D₁₀	C₅⋊₃C₂≀C₄
kernel	C₅⋊₃C₂≀C₄	C₂₀.D₄	C₅×C₂³⋊C₄	C₂³⋊D₁₀	C₂³.D₅	C₂³×D₅	C₂×C₂₀	C₂²×C₁₀	C₂³⋊C₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	1	1	2	2	4	4	4	1	2	4	2

Matrix representation of C₅⋊₃C₂≀C₄ ►in GL₆(𝔽₄₁)

35	36	0	0	0	0
40	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

6	6	0	0	0	0
1	35	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	18	1	2
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	18	0	0
0	0	0	1	0	0
0	0	0	23	40	39
0	0	0	0	0	1

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	40	0
0	0	0	23	0	40

1	0	0	0	0	0
0	1	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

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	40	0
0	0	0	1	0	32
0	0	40	0	0	0
0	0	1	23	0	40

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,36,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,18,0,0,0,0,0,1,0,0,0,0,0,2,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,18,1,23,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,1,0,0,0,1,0,23,0,0,40,0,0,0,0,0,0,32,0,40] >;

C₅⋊₃C₂≀C₄ in GAP, Magma, Sage, TeX

C_5\rtimes_3C_2\wr C_4

% in TeX

G:=Group("C5:3C2wrC4");

// GroupNames label

G:=SmallGroup(320,29);

// by ID

G=gap.SmallGroup(320,29);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,346,297,851,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;

// generators/relations

G = C5⋊3C2≀C4 order 320 = 26·5

The semidirect product of C5 and C2≀C4 acting via C2≀C4/C23⋊C4=C2

G = C₅⋊₃C₂≀C₄ order 320 = 2⁶·5

The semidirect product of C₅ and C₂≀C₄ acting via C₂≀C₄/C₂³⋊C₄=C₂