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^[×5], C₄^[×3], C₂², C₂²^[×12], C₅, C₈, C₂×C₄, C₂×C₄^[×2], D₄^[×3], C₂³^[×2], C₂³^[×4], C₁₀, C₁₀^[×5], C₂²⋊C₄, C₂²⋊C₄^[×2], M₄(2), C₂×D₄, C₂×D₄, C₂⁴, C₂₀^[×3], C₂×C₁₀, C₂×C₁₀^[×12], C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₄₀, C₂×C₂₀, C₂×C₂₀^[×2], C₅×D₄^[×3], C₂²×C₁₀^[×2], C₂²×C₁₀^[×4], C₂≀C₄, C₅×C₂²⋊C₄, C₅×C₂²⋊C₄^[×2], C₅×M₄(2), D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₅×C₂³⋊C₄, C₅×C₄.D₄, C₅×C₂²≀C₂, C₅×C₂≀C₄
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₅, C₂×C₄, D₄^[×2], C₁₀^[×3], C₂²⋊C₄, C₂₀^[×2], C₂×C₁₀, C₂³⋊C₄, C₂×C₂₀, C₅×D₄^[×2], 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 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(26 31)(27 32)(28 33)(29 34)(30 35)

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

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

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

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

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

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

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