C2^3.461C2^4 - GroupNames

G = C₂³.₄₆₁C₂⁴ order 128 = 2⁷

178^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₄₉C₂², C₂⁴.₅₈₂C₂³, C₂³.₄₆₁C₂⁴, C₂².₂₄₆2₊¹⁺⁴, (C₂×C₄²)⋊₇C₂², C₂³⋊Q₈⋊₁₉C₂, (C₂²×C₄).₃₉₂D₄, C₂³.₆₁₉(C₂×D₄), (C₂²×Q₈)⋊₅C₂², C₂⁴⋊₃C₄.₁₁C₂, (C₂²×C₄).₉₉C₂³, C₂³.₁₅₆(C₄○D₄), C₂³.₁₁D₄⋊₄₆C₂, C₂³.₃₄D₄⋊₃₆C₂, C₂.₁₄(C₂³⋊₃D₄), (C₂³×C₄).₄₀₂C₂², C₂².₃₁₂(C₂²×D₄), C₂⁴.C₂²⋊₈₇C₂, C₂.C₄²⋊₂₇C₂², C₂².₃₀(C₄.₄D₄), C₂.₆₂(C₂².₁₉C₂⁴), C₂.₅₀(C₂².₄₅C₂⁴), (C₂×C₄⋊C₄)⋊₂₃C₂², (C₄×C₂²⋊C₄)⋊₉₁C₂, (C₂×C₄).₃₅₇(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₂₄C₂, C₂.₂₃(C₂×C₄.₄D₄), C₂².₃₃₇(C₂×C₄○D₄), (C₂²×C₂²⋊C₄).₂₅C₂, (C₂×C₂²⋊C₄).₁₈₄C₂², SmallGroup(128,1293)

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

Derived series

C₁ — C₂³ — C₂³.₄₆₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³.₄₆₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₆₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₆₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₆₁C₂⁴

Generators and relations for C₂³.₄₆₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=cb=bc, e²=ca=ac, ab=ba, ede^-1=gdg=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 740 in 336 conjugacy classes, 104 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂⁵, C₄×C₂²⋊C₄, C₂⁴⋊₃C₄, C₂³.₃₄D₄, C₂⁴.C₂², C₂³⋊Q₈, C₂³.₁₁D₄, C₂²×C₂²⋊C₄, C₂×C₂²⋊Q₈, C₂³.₄₆₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂×C₄.₄D₄, C₂³⋊₃D₄, C₂².₄₅C₂⁴, C₂³.₄₆₁C₂⁴

Smallest permutation representation of C₂³.₄₆₁C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

(1 5)(3 7)(9 11)(10 30)(12 32)(13 23)(15 21)(17 19)(18 26)(20 28)(25 27)(29 31)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	4E	···	4R	4S	4T	4U	4V
order	1	2	···	2	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	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

C₂³.₄₆₁C₂⁴

C₄×C₂²⋊C₄

C₂⁴⋊₃C₄

C₂³.₃₄D₄

C₂⁴.C₂²

C₂³⋊Q₈

C₂³.₁₁D₄

C₂²×C₂²⋊C₄

C₂×C₂²⋊Q₈

C₂²×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₆₁C₂⁴ ►in GL₆(𝔽₅)

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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	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

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

2	0	0	0	0	0
0	3	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	4	0

1	0	0	0	0	0
0	1	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

1	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	4	0
0	0	0	0	0	4

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

C₂³.₄₆₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{461}C_2^4

% in TeX

G:=Group("C2^3.461C2^4");

// GroupNames label

G:=SmallGroup(128,1293);

// by ID

G=gap.SmallGroup(128,1293);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c*b=b*c,e^2=c*a=a*c,a*b=b*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,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 = C23.461C24 order 128 = 27

178th central stem extension by C23 of C24

G = C₂³.₄₆₁C₂⁴ order 128 = 2⁷

178^th central stem extension by C₂³ of C₂⁴