C2xC2^2.45C2^4 - GroupNames

G = C₂×C₂².₄₅C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₄₅C₂⁴

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂×C₂².₄₅C₂⁴, C₄²⋊₁₄C₂³, C₂².₅₈C₂⁵, C₂⁵.₇₇C₂², C₂⁴.₄₉₅C₂³, C₂³.₂₇₅C₂⁴, C₂².₁₁₁2₊¹⁺⁴, C₄⋊C₄⋊₂₀C₂³, (C₂×Q₈)⋊₇C₂³, C₂²⋊C₄⋊₈C₂³, (C₂×C₄).₅₈C₂⁴, (C₄×D₄)⋊₁₀₆C₂², (C₂³×C₄)⋊₁₇C₂², (C₂×C₄²)⋊₅₃C₂², (C₂²×C₄)⋊₁₀C₂³, C₂²⋊Q₈⋊₈₇C₂², (C₂×D₄).₄₅₄C₂³, C₄.₄D₄⋊₇₃C₂², (C₂²×Q₈)⋊₂₈C₂², C₄²⋊C₂⋊₉₄C₂², C₄²⋊₂C₂⋊₂₈C₂², C₂²≀C₂.₂₃C₂², C₂³.₃₁₉(C₄○D₄), C₂.₁₈(C₂×2₊¹⁺⁴), (C₂²×D₄).₅₉₀C₂², C₂².D₄⋊₄₃C₂², (C₂×C₄×D₄)⋊₈₄C₂, (C₂×C₂²⋊Q₈)⋊₇₁C₂, (C₂×C₄⋊C₄)⋊₁₃₅C₂², (C₂×C₄.₄D₄)⋊₅₁C₂, C₂.₃₀(C₂²×C₄○D₄), C₂².₂₄(C₂×C₄○D₄), (C₂×C₄²⋊C₂)⋊₅₉C₂, (C₂×C₄²⋊₂C₂)⋊₃₄C₂, (C₂×C₂²≀C₂).₁₇C₂, (C₂²×C₂²⋊C₄)⋊₃₃C₂, (C₂×C₂²⋊C₄)⋊₈₈C₂², (C₂×C₂².D₄)⋊₅₆C₂, SmallGroup(128,2201)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂² — C₂×C₂².₄₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂×C₂².₄₅C₂⁴

Lower central

C₁ — C₂² — C₂×C₂².₄₅C₂⁴

Upper central

C₁ — C₂³ — C₂×C₂².₄₅C₂⁴

Jennings

C₁ — C₂² — C₂×C₂².₄₅C₂⁴

Generators and relations for C₂×C₂².₄₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=b, e²=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede^-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 1084 in 664 conjugacy classes, 404 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂⁵, C₂²×C₂²⋊C₄, C₂×C₄²⋊C₂, C₂×C₄×D₄, C₂×C₂²≀C₂, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂×C₄²⋊₂C₂, C₂².₄₅C₂⁴, C₂×C₂².₄₅C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂².₄₅C₂⁴, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂×C₂².₄₅C₂⁴

Smallest permutation representation of C₂×C₂².₄₅C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

(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)

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

(2 12)(4 10)(5 26)(7 28)(14 21)(16 23)(18 30)(20 32)

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

G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,11)(2,12)(3,9)(4,10)(5,26)(6,27)(7,28)(8,25)(13,24)(14,21)(15,22)(16,23)(17,29)(18,30)(19,31)(20,32), (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), (1,31,11,19)(2,30,12,18)(3,29,9,17)(4,32,10,20)(5,23,26,16)(6,22,27,15)(7,21,28,14)(8,24,25,13), (2,12)(4,10)(5,26)(7,28)(14,21)(16,23)(18,30)(20,32), (1,11)(2,12)(3,9)(4,10)(5,26)(6,27)(7,28)(8,25)(13,22)(14,23)(15,24)(16,21)(17,31)(18,32)(19,29)(20,30)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,11)(2,12)(3,9)(4,10)(5,26)(6,27)(7,28)(8,25)(13,24)(14,21)(15,22)(16,23)(17,29)(18,30)(19,31)(20,32), (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), (1,31,11,19)(2,30,12,18)(3,29,9,17)(4,32,10,20)(5,23,26,16)(6,22,27,15)(7,21,28,14)(8,24,25,13), (2,12)(4,10)(5,26)(7,28)(14,21)(16,23)(18,30)(20,32), (1,11)(2,12)(3,9)(4,10)(5,26)(6,27)(7,28)(8,25)(13,22)(14,23)(15,24)(16,21)(17,31)(18,32)(19,29)(20,30) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,12),(6,9),(7,10),(8,11),(13,31),(14,32),(15,29),(16,30),(17,22),(18,23),(19,24),(20,21)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,11),(2,12),(3,9),(4,10),(5,26),(6,27),(7,28),(8,25),(13,24),(14,21),(15,22),(16,23),(17,29),(18,30),(19,31),(20,32)], [(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)], [(1,31,11,19),(2,30,12,18),(3,29,9,17),(4,32,10,20),(5,23,26,16),(6,22,27,15),(7,21,28,14),(8,24,25,13)], [(2,12),(4,10),(5,26),(7,28),(14,21),(16,23),(18,30),(20,32)], [(1,11),(2,12),(3,9),(4,10),(5,26),(6,27),(7,28),(8,25),(13,22),(14,23),(15,24),(16,21),(17,31),(18,32),(19,29),(20,30)]])

50 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	2Q	2R	2S	4A	···	4P	4Q	···	4AD
order	1	2	···	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	4	4	4	4	2	···	2	4	···	4

50 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂×C₂².₄₅C₂⁴

C₂²×C₂²⋊C₄

C₂×C₄²⋊C₂

C₂×C₄×D₄

C₂×C₂²≀C₂

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂×C₄.₄D₄

C₂×C₄²⋊₂C₂

C₂².₄₅C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂×C₂².₄₅C₂⁴ ►in GL₅(𝔽₅)

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	3	0
0	0	0	4	2

4	0	0	0	0
0	2	0	0	0
0	0	2	0	0
0	0	0	2	2
0	0	0	1	3

1	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	1	0
0	0	0	0	1

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	1	0
0	0	0	3	4

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,2],[4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,1,0,0,0,2,3],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,3,0,0,0,0,4] >;

C₂×C₂².₄₅C₂⁴ in GAP, Magma, Sage, TeX

C_2\times C_2^2._{45}C_2^4

% in TeX

G:=Group("C2xC2^2.45C2^4");

// GroupNames label

G:=SmallGroup(128,2201);

// by ID

G=gap.SmallGroup(128,2201);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,570]);

// Polycyclic

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

// generators/relations

G = C2×C22.45C24 order 128 = 27

Direct product of C2 and C22.45C24

G = C₂×C₂².₄₅C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₄₅C₂⁴