C5xC2wrC2^2 - GroupNames

G = C₅×C₂≀C₂² order 320 = 2⁶·5

Direct product of C₅ and C₂≀C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₅×C₂≀C₂², 2₊¹⁺⁴⋊₂C₁₀, C₂³⋊(C₅×D₄), (C₂×C₂₀)⋊₄D₄, C₂³⋊C₄⋊₃C₁₀, C₂⁴⋊₂(C₂×C₁₀), (C₂²×C₁₀)⋊₁D₄, C₂²≀C₂⋊₁C₁₀, (C₂³×C₁₀)⋊₁C₂², C₂².₁₈(D₄×C₁₀), C₁₀.₁₀₄C₂²≀C₂, C₂³.₃(C₂²×C₁₀), (C₅×2₊¹⁺⁴)⋊₈C₂, (D₄×C₁₀).₁₈₃C₂², (C₂²×C₁₀).₈₂C₂³, (C₂×C₄)⋊(C₅×D₄), (C₅×C₂³⋊C₄)⋊₉C₂, C₂²⋊C₄⋊₁(C₂×C₁₀), (C₂×D₄).₈(C₂×C₁₀), (C₅×C₂²≀C₂)⋊₁₁C₂, C₂.₁₈(C₅×C₂²≀C₂), (C₂×C₁₀).₄₁₃(C₂×D₄), (C₅×C₂²⋊C₄)⋊₃₆C₂², SmallGroup(320,958)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₅×C₂≀C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₁₀ — C₂³×C₁₀ — C₅×C₂²≀C₂ — C₅×C₂≀C₂²

Lower central

C₁ — C₂ — C₂³ — C₅×C₂≀C₂²

Upper central

C₁ — C₁₀ — C₂²×C₁₀ — C₅×C₂≀C₂²

Generators and relations for C₅×C₂≀C₂²
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf=bcd, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂≀C₂², C₅×C₂²⋊C₄, C₅×C₂²⋊C₄, D₄×C₁₀, D₄×C₁₀, C₅×C₄○D₄, C₂³×C₁₀, C₅×C₂³⋊C₄, C₅×C₂²≀C₂, C₅×2₊¹⁺⁴, C₅×C₂≀C₂²
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂×C₁₀, C₂²≀C₂, C₅×D₄, C₂²×C₁₀, C₂≀C₂², D₄×C₁₀, C₅×C₂²≀C₂, C₅×C₂≀C₂²

Smallest permutation representation of C₅×C₂≀C₂²
►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 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 31)(12 32)(13 33)(14 34)(15 35)(16 26)(17 27)(18 28)(19 29)(20 30)

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	···	2I	4A	4B	4C	4D	4E	4F	5A	5B	5C	5D	10A	10B	10C	10D	10E	···	10P	10Q	···	10AJ	20A	···	20L	20M	···	20X
order	1	2	2	2	2	2	···	2	4	4	4	4	4	4	5	5	5	5	10	10	10	10	10	···	10	10	···	10	20	···	20	20	···	20
size	1	1	2	2	2	4	···	4	4	4	4	8	8	8	1	1	1	1	1	1	1	1	2	···	2	4	···	4	4	···	4	8	···	8

80 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+					+	+			+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	D₄	C₅×D₄	C₅×D₄	C₂≀C₂²	C₅×C₂≀C₂²
kernel	C₅×C₂≀C₂²	C₅×C₂³⋊C₄	C₅×C₂²≀C₂	C₅×2₊¹⁺⁴	C₂≀C₂²	C₂³⋊C₄	C₂²≀C₂	2₊¹⁺⁴	C₂×C₂₀	C₂²×C₁₀	C₂×C₄	C₂³	C₅	C₁
# reps	1	3	3	1	4	12	12	4	3	3	12	12	2	8

Matrix representation of C₅×C₂≀C₂² ►in GL₆(𝔽₄₁)

10	0	0	0	0	0
0	10	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	1	0
0	0	0	0	0	40
0	0	1	0	0	0
0	0	0	40	0	0

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

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

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

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₅×C₂≀C₂² in GAP, Magma, Sage, TeX

C_5\times C_2\wr C_2^2

% in TeX

G:=Group("C5xC2wrC2^2");

// GroupNames label

G:=SmallGroup(320,958);

// by ID

G=gap.SmallGroup(320,958);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1768,5052]);

// Polycyclic

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

// generators/relations

G = C5×C2≀C22 order 320 = 26·5

Direct product of C5 and C2≀C22

G = C₅×C₂≀C₂² order 320 = 2⁶·5

Direct product of C₅ and C₂≀C₂²