C2^3.434C2^4 - GroupNames

G = C₂³.₄₃₄C₂⁴ order 128 = 2⁷

151^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₄₇C₂², C₂³.₄₃₄C₂⁴, C₂⁴.₃₁₇C₂³, C₂².₂₂₄2₊¹⁺⁴, (C₂²×C₄)⋊₂₈D₄, (C₂×D₄).₂₁₁D₄, C₂³.₄₅(C₂×D₄), C₂⁴⋊₃C₄⋊₁₈C₂, (C₂³×C₄)⋊₁₁C₂², (C₂×C₄²)⋊₂₅C₂², C₂.₆₅(D₄⋊₅D₄), C₂³.Q₈⋊₂₆C₂, C₂³.₇Q₈⋊₆₂C₂, C₂³.₁₅₀(C₄○D₄), C₂³.₂₃D₄⋊₅₂C₂, C₂³.₃₄D₄⋊₃₅C₂, C₂³.₁₀D₄⋊₃₈C₂, C₂.₁₁(C₂³⋊₃D₄), C₂².₇₆(C₄⋊D₄), (C₂²×C₄).₅₃₅C₂³, C₂².₂₈₅(C₂²×D₄), C₂⁴.C₂²⋊₇₆C₂, C₂.C₄²⋊₂₆C₂², (C₂²×D₄).₁₆₀C₂², C₂.₅₇(C₂².₁₉C₂⁴), C₂.₄₅(C₂².₄₅C₂⁴), (C₂×C₄×D₄)⋊₄₁C₂, (C₂×C₄⋊C₄)⋊₂₁C₂², (C₂×C₄).₃₅₀(C₂×D₄), C₂.₂₉(C₂×C₄⋊D₄), (C₂×C₂²≀C₂).₁₁C₂, (C₂×C₂²⋊C₄)⋊₂₀C₂², (C₂²×C₂²⋊C₄)⋊₂₃C₂, C₂².₃₁₁(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₁₉C₂, SmallGroup(128,1266)

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

Derived series

C₁ — C₂³ — C₂³.₄₃₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³.₄₃₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₃₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₃₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₃₄C₂⁴

Generators and relations for C₂³.₄₃₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=g²=1, e²=ca=ac, ab=ba, 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: 884 in 400 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, 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×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂²×C₂²⋊C₄, C₂×C₄×D₄, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂³.₄₃₄C₂⁴
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, C₂³⋊₃D₄, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂³.₄₃₄C₂⁴

Smallest permutation representation of C₂³.₄₃₄C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

(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 19)(2 21)(3 17)(4 23)(6 32)(8 30)(9 16)(11 14)(18 25)(20 27)(22 28)(24 26)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2Q	4A	4B	4C	4D	4E	···	4P	4Q	4R	4S	4T
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

C₂³.₄₃₄C₂⁴

C₂⁴⋊₃C₄

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂²×C₂²⋊C₄

C₂×C₄×D₄

C₂×C₂²≀C₂

C₂×C₂².D₄

C₂²×C₄

C₂×D₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₃₄C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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	4	1

4	0	0	0	0	0
0	4	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

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

C₂³.₄₃₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{434}C_2^4

% in TeX

G:=Group("C2^3.434C2^4");

// GroupNames label

G:=SmallGroup(128,1266);

// by ID

G=gap.SmallGroup(128,1266);

# by ID

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

// 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=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,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 = C23.434C24 order 128 = 27

151st central stem extension by C23 of C24

G = C₂³.₄₃₄C₂⁴ order 128 = 2⁷

151^st central stem extension by C₂³ of C₂⁴