C2^3.304C2^4 - GroupNames

G = C₂³.₃₀₄C₂⁴ order 128 = 2⁷

21^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₃₂C₂², C₂⁴.₂₄₂C₂³, C₂³.₃₀₄C₂⁴, C₂².₁₂₂2₊¹⁺⁴, (C₂×D₄)⋊₄₁D₄, (C₂²×C₄)⋊₁₉D₄, C₂³⋊₅(C₄○D₄), C₂³⋊₂D₄⋊₄C₂, C₂³⋊Q₈⋊₃C₂, (C₂³×C₄)⋊₁₉C₂², C₂³.₁₄₇(C₂×D₄), C₂.₁₂(D₄⋊₅D₄), C₂².₁₄C₂²≀C₂, (C₂²×D₄)⋊₅₄C₂², (C₂²×Q₈)⋊₅₂C₂², C₂³.₂₃D₄⋊₂₁C₂, C₂³.₃₄D₄⋊₁₇C₂, (C₂²×C₄).₇₈₆C₂³, C₂².₁₈₄(C₂²×D₄), C₂.C₄²⋊₁₅C₂², C₂.₈(C₂².₂₉C₂⁴), C₂.₁₆(C₂².₁₉C₂⁴), (C₂×C₂²≀C₂)⋊₃C₂, (C₂×C₄).₃₀₀(C₂×D₄), (C₂²×C₄○D₄)⋊₂C₂, C₂.₁₁(C₂×C₂²≀C₂), (C₂×C₂²⋊C₄)⋊₉C₂², (C₂²×C₂²⋊C₄)⋊₁₆C₂, C₂².₁₈₃(C₂×C₄○D₄), SmallGroup(128,1136)

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

Derived series

C₁ — C₂³ — C₂³.₃₀₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³.₃₀₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₀₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₀₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₀₄C₂⁴

Generators and relations for C₂³.₃₀₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=g²=1, e²=a, ab=ba, ac=ca, ede^-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1140 in 528 conjugacy classes, 120 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂⁵, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³⋊₂D₄, C₂³⋊Q₈, C₂²×C₂²⋊C₄, C₂×C₂²≀C₂, C₂²×C₄○D₄, C₂³.₃₀₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂²≀C₂, C₂².₁₉C₂⁴, C₂².₂₉C₂⁴, D₄⋊₅D₄, C₂³.₃₀₄C₂⁴

Smallest permutation representation of C₂³.₃₀₄C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2S	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R
order	1	2	···	2	2	2	2	2	2	···	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	4	···	4	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₀₄C₂⁴

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³⋊₂D₄

C₂³⋊Q₈

C₂²×C₂²⋊C₄

C₂×C₂²≀C₂

C₂²×C₄○D₄

C₂²×C₄

C₂×D₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₀₄C₂⁴ ►in GL₆(𝔽₅)

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

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

1	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	4	0
0	0	0	0	0	4

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

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

4	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	4	0
0	0	0	0	0	1

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂³.₃₀₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{304}C_2^4

% in TeX

G:=Group("C2^3.304C2^4");

// GroupNames label

G:=SmallGroup(128,1136);

// by ID

G=gap.SmallGroup(128,1136);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675]);

// Polycyclic

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

// generators/relations

G = C23.304C24 order 128 = 27

21st central stem extension by C23 of C24

G = C₂³.₃₀₄C₂⁴ order 128 = 2⁷

21^st central stem extension by C₂³ of C₂⁴