C2^3.570C2^4 - GroupNames

G = C₂³.₅₇₀C₂⁴ order 128 = 2⁷

287^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₅₆C₂², C₂³.₅₇₀C₂⁴, C₂⁴.₃₈₃C₂³, C₂².₃₄₄2₊¹⁺⁴, C₂.₃₄D₄², C₂²⋊C₄⋊₉D₄, C₂⁴⋊₃C₄⋊₂₂C₂, C₂³⋊₂D₄⋊₃₃C₂, (C₂×C₄²)⋊₂₈C₂², C₂³⋊Q₈⋊₃₇C₂, C₂³.₂₀₂(C₂×D₄), C₂.₈₂(D₄⋊₅D₄), (C₂²×Q₈)⋊₇C₂², (C₂²×D₄)⋊₁₂C₂², C₂³.₁₆₆(C₄○D₄), C₂³.₁₀D₄⋊₇₀C₂, (C₂²×C₄).₁₇₅C₂³, C₂².₃₇₉(C₂²×D₄), C₂.C₄²⋊₃₄C₂², C₂⁴.₃C₂²⋊₇₁C₂, C₂.₃(C₂⁴⋊C₂²), C₂⁴.C₂²⋊₁₁₅C₂, C₂.₅₇(C₂².₃₂C₂⁴), C₂.₅₂(C₂².₂₉C₂⁴), (C₂×C₄⋊C₄)⋊₃₀C₂², (C₂×C₄).₄₁₀(C₂×D₄), (C₂×C₂²≀C₂)⋊₁₃C₂, (C₂×C₄.₄D₄)⋊₂₄C₂, (C₂×C₂²⋊C₄)⋊₂₇C₂², C₂².₄₃₆(C₂×C₄○D₄), SmallGroup(128,1402)

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

Derived series

C₁ — C₂³ — C₂³.₅₇₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂×C₂²≀C₂ — C₂³.₅₇₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₇₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₇₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₇₀C₂⁴

Generators and relations for C₂³.₅₇₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=e²=a, 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: 980 in 398 conjugacy classes, 104 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄.₄D₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂⁵, C₂⁴⋊₃C₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³⋊₂D₄, C₂³⋊Q₈, C₂³.₁₀D₄, C₂×C₂²≀C₂, C₂×C₄.₄D₄, C₂³.₅₇₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, D₄², D₄⋊₅D₄, C₂⁴⋊C₂², C₂³.₅₇₀C₂⁴

Smallest permutation representation of C₂³.₅₇₀C₂⁴
►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 20 3 18)(2 19 4 17)(5 21 7 23)(6 24 8 22)(9 31 11 29)(10 30 12 32)(13 25 15 27)(14 28 16 26)

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₇₀C₂⁴

C₂⁴⋊₃C₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³⋊Q₈

C₂³.₁₀D₄

C₂×C₂²≀C₂

C₂×C₄.₄D₄

C₂²⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₇₀C₂⁴ ►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	3	0	0	0	0
3	0	0	0	0	0
0	0	3	3	0	0
0	0	4	2	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
4	0	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	1	0

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

C₂³.₅₇₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{570}C_2^4

% in TeX

G:=Group("C2^3.570C2^4");

// GroupNames label

G:=SmallGroup(128,1402);

// by ID

G=gap.SmallGroup(128,1402);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=a,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 = C23.570C24 order 128 = 27

287th central stem extension by C23 of C24

G = C₂³.₅₇₀C₂⁴ order 128 = 2⁷

287^th central stem extension by C₂³ of C₂⁴