C2^3.585C2^4 - GroupNames

G = C₂³.₅₈₅C₂⁴ order 128 = 2⁷

302^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₅₉C₂², C₂⁴.₃₉₂C₂³, C₂³.₅₈₅C₂⁴, C₂².₃₅₉2₊¹⁺⁴, C₂²⋊C₄.₁₁D₄, C₂⁴⋊₃C₄⋊₂₅C₂, (C₂×C₄²)⋊₃₂C₂², C₂³.₂₀₉(C₂×D₄), C₂.₉₀(D₄⋊₅D₄), C₂³.₄Q₈⋊₄₂C₂, C₂³.Q₈⋊₅₅C₂, C₂³.₁₆₉(C₄○D₄), C₂³.₁₀D₄⋊₈₀C₂, (C₂²×C₄).₁₈₀C₂³, C₂².₃₉₄(C₂²×D₄), C₂.C₄²⋊₃₈C₂², C₂⁴.₃C₂²⋊₇₇C₂, (C₂²×D₄).₂₂₄C₂², C₂⁴.C₂²⋊₁₂₃C₂, C₂.₆₂(C₂².₃₂C₂⁴), C₂.₅₆(C₂².₂₉C₂⁴), C₂.₇₆(C₂².₄₅C₂⁴), C₂.₁₁(C₂².₅₄C₂⁴), (C₂×C₄).₉₂(C₂×D₄), (C₂×C₄⋊C₄)⋊₃₄C₂², (C₂×C₂²⋊C₄)⋊₆C₂², (C₂×C₄²⋊₂C₂)⋊₁₇C₂, (C₂×C₂²≀C₂).₁₄C₂, C₂².₄₄₇(C₂×C₄○D₄), SmallGroup(128,1417)

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

Derived series

C₁ — C₂³ — C₂³.₅₈₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴⋊₃C₄ — C₂³.₅₈₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₈₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₈₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₈₅C₂⁴

Generators and relations for C₂³.₅₈₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=a, e²=ba=ab, 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: 804 in 333 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄²⋊₂C₂, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₄Q₈, C₂×C₂²≀C₂, C₂×C₄²⋊₂C₂, C₂³.₅₈₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂².₅₄C₂⁴, C₂³.₅₈₅C₂⁴

Smallest permutation representation of C₂³.₅₈₅C₂⁴
►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 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(21 32)(22 29)(23 30)(24 31)

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

(1 8)(2 12)(3 6)(4 10)(5 26)(7 28)(9 27)(11 25)(13 18)(15 20)(22 29)(24 31)

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2M	2N	4A	···	4L	4M	···	4Q
order	1	2	···	2	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₈₅C₂⁴

C₂⁴⋊₃C₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₄Q₈

C₂×C₂²≀C₂

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₈₅C₂⁴ ►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	2	0	0	0	0
2	0	0	0	0	0
0	0	4	4	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
4	0	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	1	3
0	0	0	0	0	4

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

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

C₂³.₅₈₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{585}C_2^4

% in TeX

G:=Group("C2^3.585C2^4");

// GroupNames label

G:=SmallGroup(128,1417);

// by ID

G=gap.SmallGroup(128,1417);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=a,e^2=b*a=a*b,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 = C23.585C24 order 128 = 27

302nd central stem extension by C23 of C24

G = C₂³.₅₈₅C₂⁴ order 128 = 2⁷

302^nd central stem extension by C₂³ of C₂⁴