C5xC2^2wrC2 - GroupNames

G = C₅×C₂²≀C₂ order 160 = 2⁵·5

Direct product of C₅ and C₂²≀C₂

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

Aliases: C₅×C₂²≀C₂, C₂⁴⋊₂C₁₀, (C₂×C₁₀)⋊₇D₄, (C₂×D₄)⋊₁C₁₀, C₂.₄(D₄×C₁₀), C₂²⋊₂(C₅×D₄), (D₄×C₁₀)⋊₁₀C₂, C₂²⋊C₄⋊₂C₁₀, (C₂×C₂₀)⋊₈C₂², C₂³⋊₁(C₂×C₁₀), (C₂³×C₁₀)⋊₁C₂, C₁₀.₆₇(C₂×D₄), (C₂×C₁₀).₇₅C₂³, (C₂²×C₁₀)⋊₁C₂², C₂².₁₀(C₂²×C₁₀), (C₂×C₄)⋊₁(C₂×C₁₀), (C₅×C₂²⋊C₄)⋊₁₀C₂, SmallGroup(160,181)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×C₂²≀C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₅×C₂²≀C₂

Lower central

C₁ — C₂² — C₅×C₂²≀C₂

Upper central

C₁ — C₂×C₁₀ — C₅×C₂²≀C₂

Generators and relations for C₅×C₂²≀C₂
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂²≀C₂, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₅×C₂²⋊C₄, D₄×C₁₀, C₂³×C₁₀, C₅×C₂²≀C₂
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂×C₁₀, C₂²≀C₂, C₅×D₄, C₂²×C₁₀, D₄×C₁₀, C₅×C₂²≀C₂

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

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

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

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

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

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

C₅×C₂²≀C₂ is a maximal subgroup of
C₂⁴⋊₂Dic₅ C₂⁴⋊₂D₁₀ C₂⁴.₅₆D₁₀ C₂⁴.₃₂D₁₀ C₂⁴⋊₃D₁₀ C₂⁴⋊₄D₁₀ C₂⁴.₃₃D₁₀ C₂⁴.₃₄D₁₀ C₂⁴.₃₅D₁₀ C₂⁴⋊₅D₁₀ C₂⁴.₃₆D₁₀ C₅×D₄²

70 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	4A	4B	4C	5A	5B	5C	5D	10A	···	10L	10M	···	10AJ	10AK	10AL	10AM	10AN	20A	···	20L
order	1	2	2	2	2	···	2	2	4	4	4	5	5	5	5	10	···	10	10	···	10	10	10	10	10	20	···	20
size	1	1	1	1	2	···	2	4	4	4	4	1	1	1	1	1	···	1	2	···	2	4	4	4	4	4	···	4

70 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	D₄	C₅×D₄
kernel	C₅×C₂²≀C₂	C₅×C₂²⋊C₄	D₄×C₁₀	C₂³×C₁₀	C₂²≀C₂	C₂²⋊C₄	C₂×D₄	C₂⁴	C₂×C₁₀	C₂²
# reps	1	3	3	1	4	12	12	4	6	24

Matrix representation of C₅×C₂²≀C₂ ►in GL₄(𝔽₄₁) generated by

18	0	0	0
0	18	0	0
0	0	1	0
0	0	0	1

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

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

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

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

0	1	0	0
1	0	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C₅×C₂²≀C₂ in GAP, Magma, Sage, TeX

C_5\times C_2^2\wr C_2

% in TeX

G:=Group("C5xC2^2wrC2");

// GroupNames label

G:=SmallGroup(160,181);

// by ID

G=gap.SmallGroup(160,181);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514]);

// Polycyclic

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

// generators/relations

G = C5×C22≀C2 order 160 = 25·5

Direct product of C5 and C22≀C2

G = C₅×C₂²≀C₂ order 160 = 2⁵·5

Direct product of C₅ and C₂²≀C₂