C2^3.311C2^4 - GroupNames

G = C₂³.₃₁₁C₂⁴ order 128 = 2⁷

28^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₃₆C₂², C₂⁴.₂₄₈C₂³, C₂³.₃₁₁C₂⁴, C₂².₁₂₇2₊¹⁺⁴, (C₂²×C₄)⋊₂₂D₄, C₂⁴⋊₃C₄⋊₁₄C₂, (C₂×C₄²)⋊₁₉C₂², C₂²⋊C₄.₁₂₃D₄, C₂³.₄₂₁(C₂×D₄), C₂.₁₇(D₄⋊₅D₄), C₂³.Q₈⋊₁C₂, C₂³.₁₀D₄⋊₈C₂, C₂³.₈Q₈⋊₂₇C₂, C₂³.₁₃₈(C₄○D₄), C₂³.₂₃D₄⋊₂₆C₂, C₂².₇₂(C₄⋊D₄), (C₂²×C₄).₅₀₃C₂³, (C₂³×C₄).₃₂₉C₂², C₂².₁₉₁(C₂²×D₄), C₂⁴.₃C₂²⋊₂₇C₂, C₂⁴.C₂²⋊₂₉C₂, C₂.C₄²⋊₁₉C₂², (C₂²×D₄).₁₁₇C₂², C₂.₁₀(C₂².₂₉C₂⁴), C₂.₁₉(C₂².₁₉C₂⁴), C₂.₁₀(C₂².₄₅C₂⁴), (C₂×C₄⋊C₄)⋊₁₂C₂², (C₂×C₄).₃₀₅(C₂×D₄), C₂.₁₅(C₂×C₄⋊D₄), (C₂×C₂²≀C₂).₇C₂, (C₂×C₄²⋊C₂)⋊₁₈C₂, (C₂×C₂²⋊C₄)⋊₁₃C₂², (C₂²×C₂²⋊C₄)⋊₁₈C₂, C₂².₁₉₀(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄C₂, SmallGroup(128,1143)

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

Derived series

C₁ — C₂³ — C₂³.₃₁₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³.₃₁₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₁₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₁₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₁₁C₂⁴

Generators and relations for C₂³.₃₁₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=b, 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: 868 in 393 conjugacy classes, 112 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²≀C₂, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂²×C₂²⋊C₄, C₂×C₄²⋊C₂, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂³.₃₁₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄⋊D₄, C₂².₁₉C₂⁴, C₂².₂₉C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂³.₃₁₁C₂⁴

Smallest permutation representation of C₂³.₃₁₁C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

(1 3)(5 18)(6 17)(7 20)(8 19)(9 31)(10 30)(11 29)(12 32)(13 15)(21 23)(25 27)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₁₁C₂⁴

C₂⁴⋊₃C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂²×C₂²⋊C₄

C₂×C₄²⋊C₂

C₂×C₂²≀C₂

C₂×C₂².D₄

C₂²⋊C₄

C₂²×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₁₁C₂⁴ ►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	1	0
0	0	0	0	0	1

2	0	0	0	0	0
0	3	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	3
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	1	0	0
0	0	0	0	1	0
0	0	0	0	4	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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[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] >;

C₂³.₃₁₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{311}C_2^4

% in TeX

G:=Group("C2^3.311C2^4");

// GroupNames label

G:=SmallGroup(128,1143);

// by ID

G=gap.SmallGroup(128,1143);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=b,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 = C23.311C24 order 128 = 27

28th central stem extension by C23 of C24

G = C₂³.₃₁₁C₂⁴ order 128 = 2⁷

28^th central stem extension by C₂³ of C₂⁴