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^[×8], C₄^[×6], C₂²^[×3], C₂²^[×18], C₅, C₂×C₄^[×3], C₂×C₄^[×6], D₄^[×15], Q₈, C₂³, C₂³^[×3], C₂³^[×6], C₁₀, C₁₀^[×8], C₂²⋊C₄^[×3], C₂²⋊C₄^[×3], C₂×D₄^[×3], C₂×D₄^[×6], C₄○D₄^[×3], C₂⁴, C₂₀^[×6], C₂×C₁₀^[×3], C₂×C₁₀^[×18], C₂³⋊C₄^[×3], C₂²≀C₂^[×3], 2₊¹⁺⁴, C₂×C₂₀^[×3], C₂×C₂₀^[×6], C₅×D₄^[×15], C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀^[×3], C₂²×C₁₀^[×6], C₂≀C₂², C₅×C₂²⋊C₄^[×3], C₅×C₂²⋊C₄^[×3], D₄×C₁₀^[×3], D₄×C₁₀^[×6], C₅×C₄○D₄^[×3], C₂³×C₁₀, C₅×C₂³⋊C₄^[×3], C₅×C₂²≀C₂^[×3], C₅×2₊¹⁺⁴, C₅×C₂≀C₂²
Quotients: C₁, C₂^[×7], C₂²^[×7], C₅, D₄^[×6], C₂³, C₁₀^[×7], C₂×D₄^[×3], C₂×C₁₀^[×7], C₂²≀C₂, C₅×D₄^[×6], C₂²×C₁₀, C₂≀C₂², D₄×C₁₀^[×3], 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 26)(7 27)(8 28)(9 29)(10 30)(11 22)(12 23)(13 24)(14 25)(15 21)(16 35)(17 31)(18 32)(19 33)(20 34)

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

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

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

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

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

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

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

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