C2^3.308C2^4 - GroupNames

G = C₂³.₃₀₈C₂⁴ order 128 = 2⁷

25^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₃₄C₂², C₂⁴.₂₄₅C₂³, C₂³.₃₀₈C₂⁴, C₂².₁₂₅2₊¹⁺⁴, C₂.₇D₄², C₄⋊₂C₂²≀C₂, (C₂×D₄)⋊₂₉D₄, C₂³⋊₂(C₂×D₄), (D₄×C₂³)⋊₃C₂, (C₂²×C₄)⋊₂₁D₄, C₂³⋊₂D₄⋊₆C₂, C₂⁴⋊₃C₄⋊₁₂C₂, C₂²⋊₄(C₄⋊D₄), (C₂³×C₄)⋊₂₀C₂², (C₂×C₄²)⋊₁₈C₂², (C₂²×D₄)⋊₃C₂², C₂.₁₅(D₄⋊₅D₄), C₂³.₁₀D₄⋊₆C₂, C₂³.₇Q₈⋊₃₁C₂, C₂³.₂₉₇(C₄○D₄), (C₂²×C₄).₇₈₉C₂³, C₂².₁₈₈(C₂²×D₄), C₂.C₄²⋊₁₇C₂², C₂⁴.₃C₂²⋊₂₆C₂, C₂.₉(C₂².₂₉C₂⁴), (C₂×C₄×D₄)⋊₂₀C₂, (C₂×C₄)⋊₂(C₂×D₄), (C₂×C₄⋊D₄)⋊₃C₂, (C₂×C₂²≀C₂)⋊₄C₂, C₂.₁₃(C₂×C₄⋊D₄), (C₂×C₄⋊C₄)⋊₁₀₉C₂², C₂.₁₅(C₂×C₂²≀C₂), (C₂×C₂²⋊C₄)⋊₁₁C₂², C₂².₁₈₇(C₂×C₄○D₄), SmallGroup(128,1140)

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

Derived series

C₁ — C₂³ — C₂³.₃₀₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂³.₃₀₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₀₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₀₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₀₈C₂⁴

Generators and relations for C₂³.₃₀₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=g²=a, ab=ba, ac=ca, ede^-1=gdg^-1=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: 1396 in 616 conjugacy classes, 132 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₇Q₈, C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂×C₄×D₄, C₂×C₂²≀C₂, C₂×C₄⋊D₄, D₄×C₂³, C₂³.₃₀₈C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂²≀C₂, C₂×C₄⋊D₄, C₂².₂₉C₂⁴, D₄², D₄⋊₅D₄, C₂³.₃₀₈C₂⁴

Smallest permutation representation of C₂³.₃₀₈C₂⁴
►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 20)(2 17)(3 18)(4 19)(5 13)(6 14)(7 15)(8 16)(9 32)(10 29)(11 30)(12 31)(21 26)(22 27)(23 28)(24 25)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2U	4A	4B	4C	4D	4E	···	4L	4M	4N	4O	4P
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

C₂³.₃₀₈C₂⁴

C₂⁴⋊₃C₄

C₂³.₇Q₈

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂×C₄×D₄

C₂×C₂²≀C₂

C₂×C₄⋊D₄

D₄×C₂³

C₂²×C₄

C₂×D₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₀₈C₂⁴ ►in GL₆(ℤ)

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	-1	0
0	0	0	0	0	-1

-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	1	0
0	0	0	0	0	1

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	1	0
0	0	0	0	0	1

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	0	1
0	0	0	0	1	0

-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	0	1
0	0	0	0	-1	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	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	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,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;

C₂³.₃₀₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{308}C_2^4

% in TeX

G:=Group("C2^3.308C2^4");

// GroupNames label

G:=SmallGroup(128,1140);

// by ID

G=gap.SmallGroup(128,1140);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=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 = C23.308C24 order 128 = 27

25th central stem extension by C23 of C24

G = C₂³.₃₀₈C₂⁴ order 128 = 2⁷

25^th central stem extension by C₂³ of C₂⁴