C5xC2wrC4 - GroupNames

G = C₅×C₂≀C₄ order 320 = 2⁶·5

Direct product of C₅ and C₂≀C₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₅×C₂≀C₄, C₂⁴⋊₂C₂₀, C₂³⋊C₄⋊₁C₁₀, C₂²⋊C₄⋊₁C₂₀, (C₂³×C₁₀)⋊₁C₄, (C₂×C₂₀).₁₇D₄, C₂³.₁(C₅×D₄), C₄.D₄⋊₅C₁₀, C₂³.₁(C₂×C₂₀), (C₂²×C₁₀).₁D₄, C₂²≀C₂.₁C₁₀, C₁₀.₅₃(C₂³⋊C₄), (D₄×C₁₀).₁₇₄C₂², (C₂×C₄).₁(C₅×D₄), (C₅×C₂³⋊C₄)⋊₇C₂, (C₅×C₂²⋊C₄)⋊₉C₄, C₂.₆(C₅×C₂³⋊C₄), (C₂×D₄).₁(C₂×C₁₀), (C₅×C₂²≀C₂).₃C₂, (C₅×C₄.D₄)⋊₁₂C₂, (C₂²×C₁₀).₃₃(C₂×C₄), C₂².₁₀(C₅×C₂²⋊C₄), (C₂×C₁₀).₁₃₇(C₂²⋊C₄), SmallGroup(320,156)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₅×C₂≀C₄

Chief series

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

Lower central

C₁ — C₂ — C₂² — C₂³ — C₅×C₂≀C₄

Upper central

C₁ — C₁₀ — C₂×C₁₀ — D₄×C₁₀ — C₅×C₂≀C₄

Generators and relations for C₅×C₂≀C₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f⁴=1, ab=ba, 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: 258 in 94 conjugacy classes, 26 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₄₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂≀C₄, C₅×C₂²⋊C₄, C₅×C₂²⋊C₄, C₅×M₄(2), D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₅×C₂³⋊C₄, C₅×C₄.D₄, C₅×C₂²≀C₂, C₅×C₂≀C₄
Quotients: C₁, C₂, C₄, C₂², C₅, C₂×C₄, D₄, C₁₀, C₂²⋊C₄, C₂₀, C₂×C₁₀, C₂³⋊C₄, C₂×C₂₀, C₅×D₄, C₂≀C₄, C₅×C₂²⋊C₄, C₅×C₂³⋊C₄, C₅×C₂≀C₄

Smallest permutation representation of C₅×C₂≀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)

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

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

(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)

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

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+	+									+	+			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₅	C₁₀	C₁₀	C₁₀	C₂₀	C₂₀	D₄	D₄	C₅×D₄	C₅×D₄	C₂³⋊C₄	C₂≀C₄	C₅×C₂³⋊C₄	C₅×C₂≀C₄
kernel	C₅×C₂≀C₄	C₅×C₂³⋊C₄	C₅×C₄.D₄	C₅×C₂²≀C₂	C₅×C₂²⋊C₄	C₂³×C₁₀	C₂≀C₄	C₂³⋊C₄	C₄.D₄	C₂²≀C₂	C₂²⋊C₄	C₂⁴	C₂×C₂₀	C₂²×C₁₀	C₂×C₄	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	4	4	4	4	8	8	1	1	4	4	1	2	4	8

Matrix representation of C₅×C₂≀C₄ ►in GL₄(𝔽₄₁) generated by

37	0	0	0
0	37	0	0
0	0	37	0
0	0	0	37

40	0	0	0
0	40	0	0
0	0	0	40
0	0	40	0

1	39	0	0
0	40	0	0
0	1	0	1
0	1	1	0

1	0	0	0
0	1	0	0
40	0	40	0
40	0	0	40

40	0	0	0
0	40	0	0
0	0	40	0
0	0	0	40

40	0	39	0
0	0	40	1
0	0	1	0
1	40	1	0

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

C₅×C₂≀C₄ in GAP, Magma, Sage, TeX

C_5\times C_2\wr C_4

% in TeX

G:=Group("C5xC2wrC4");

// GroupNames label

G:=SmallGroup(320,156);

// by ID

G=gap.SmallGroup(320,156);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2111,375,10085]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,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×C2≀C4 order 320 = 26·5

Direct product of C5 and C2≀C4

G = C₅×C₂≀C₄ order 320 = 2⁶·5

Direct product of C₅ and C₂≀C₄