C2^3.584C2^4 - GroupNames

G = C₂³.₅₈₄C₂⁴ order 128 = 2⁷

301^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₅₈C₂², C₂³.₅₈₄C₂⁴, C₂⁴.₃₉₁C₂³, C₂².₃₅₈2₊¹⁺⁴, (C₂×D₄).₁₃₈D₄, C₂³.₆₃(C₂×D₄), C₂⁴⋊₃C₄⋊₂₄C₂, (C₂³×C₄)⋊₁₂C₂², (C₂×C₄²)⋊₃₁C₂², C₂³⋊Q₈⋊₄₁C₂, C₂.₈₉(D₄⋊₅D₄), (C₂²×Q₈)⋊₈C₂², C₂³.₁₆₈(C₄○D₄), C₂³.₂₃D₄⋊₈₃C₂, C₂³.₁₀D₄⋊₇₉C₂, C₂³.₁₁D₄⋊₇₇C₂, C₂.₄₂(C₂³⋊₃D₄), (C₂²×C₄).₁₇₉C₂³, C₂².₃₉₃(C₂²×D₄), C₂.C₄²⋊₃₇C₂², C₂.₅(C₂⁴⋊C₂²), (C₂²×D₄).₂₂₃C₂², C₂⁴.C₂²⋊₁₂₂C₂, C₂.₆₁(C₂².₃₂C₂⁴), C₂.₇₅(C₂².₄₅C₂⁴), (C₂×C₄⋊C₄)⋊₃₃C₂², (C₂×C₄).₄₁₆(C₂×D₄), (C₂×C₄.₄D₄)⋊₂₇C₂, (C₂×C₂²≀C₂).₁₃C₂, (C₂×C₂²⋊C₄)⋊₃₀C₂², C₂².₄₄₆(C₂×C₄○D₄), SmallGroup(128,1416)

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

Derived series

C₁ — C₂³ — C₂³.₅₈₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴⋊₃C₄ — C₂³.₅₈₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₈₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₈₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₈₄C₂⁴

Generators and relations for C₂³.₅₈₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=g²=1, e²=b, ab=ba, ac=ca, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 836 in 344 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂⁵, C₂⁴⋊₃C₄, C₂³.₂₃D₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₂²≀C₂, C₂×C₄.₄D₄, C₂³.₅₈₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂⁴⋊C₂², C₂³.₅₈₄C₂⁴

Smallest permutation representation of C₂³.₅₈₄C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

(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 13)(2 4)(3 15)(5 23)(6 8)(7 21)(10 28)(12 26)(14 16)(18 31)(20 29)(22 24)

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4J	4K	···	4P
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₈₄C₂⁴

C₂⁴⋊₃C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₂²≀C₂

C₂×C₄.₄D₄

C₂×D₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₈₄C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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	0
0	0	0	0	0	1

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	1	0
0	0	0	0	2	4

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

C₂³.₅₈₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{584}C_2^4

% in TeX

G:=Group("C2^3.584C2^4");

// GroupNames label

G:=SmallGroup(128,1416);

// by ID

G=gap.SmallGroup(128,1416);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,1571,346]);

// 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=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=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,g*d*g=a*b*d,f*g=g*f>;

// generators/relations

G = C23.584C24 order 128 = 27

301st central stem extension by C23 of C24

G = C₂³.₅₈₄C₂⁴ order 128 = 2⁷

301^st central stem extension by C₂³ of C₂⁴