C2^2.89C2^5 - GroupNames

G = C₂².₈₉C₂⁵ order 128 = 2⁷

70^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₈₉C₂⁵, C₂³.₄₅C₂⁴, C₄².₅₇₇C₂³, C₂⁴.₁₃₆C₂³, C₄○D₄⋊₁₂D₄, D₄.₆₀(C₂×D₄), D₄ ○(C₄.₄D₄), C₄⋊Q₈⋊₉₁C₂², Q₈.₆₂(C₂×D₄), Q₈ ○(C₄.₄D₄), D₄⋊₅D₄⋊₁₉C₂, Q₈⋊₅D₄⋊₁₇C₂, (C₄×D₄)⋊₄₅C₂², (C₂×C₄).₇₉C₂⁴, C₂.₃₅(D₄×C₂³), C₂²≀C₂⋊₉C₂², C₄⋊C₄.₂₉₅C₂³, C₄⋊₁D₄⋊₅₂C₂², C₄⋊D₄⋊₂₈C₂², (C₂×C₄²)⋊₆₀C₂², (C₄×Q₈)⋊₁₀₀C₂², C₄.₁₂₄(C₂²×D₄), C₂²⋊Q₈⋊₃₂C₂², (C₂×D₄).₄₇₂C₂³, C₄.₄D₄⋊₈₃C₂², C₂²⋊C₄.₂₄C₂³, (C₂×Q₈).₄₄₉C₂³, (C₂²×Q₈)⋊₃₂C₂², C₂².₁₆(C₂²×D₄), C₂².₂₉C₂⁴⋊₂₃C₂, (C₂²×C₄).₃₆₁C₂³, (C₂×2₊¹⁺⁴)⋊₁₂C₂, (C₂×2_-¹⁺⁴)⋊₁₀C₂, C₂².D₄⋊₇C₂², C₄²⋊C₂⋊₁₀₁C₂², C₂.₂₄(C₂.C₂⁵), C₂².₂₆C₂⁴⋊₃₈C₂, (C₂²×D₄).₄₂₆C₂², C₂³.₃₈C₂³⋊₂₃C₂, (C₄×C₄○D₄)⋊₂₈C₂, (C₂×C₄).₁₈₉(C₂×D₄), (C₂×C₄.₄D₄)⋊₅₄C₂, (C₂×C₄○D₄)⋊₃₁C₂², (C₂×C₂²⋊C₄)⋊₄₉C₂², SmallGroup(128,2232)

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

Derived series

C₁ — C₂² — C₂².₈₉C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₈₉C₂⁵

Lower central

C₁ — C₂² — C₂².₈₉C₂⁵

Upper central

C₁ — C₂² — C₂².₈₉C₂⁵

Jennings

C₁ — C₂² — C₂².₈₉C₂⁵

Generators and relations for C₂².₈₉C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=ba=ab, g²=a, dcd=gcg^-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece^-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1260 in 770 conjugacy classes, 428 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₄×C₄○D₄, C₂×C₄.₄D₄, C₂².₂₆C₂⁴, C₂².₂₉C₂⁴, C₂³.₃₈C₂³, D₄⋊₅D₄, Q₈⋊₅D₄, C₂×2₊¹⁺⁴, C₂×2_-¹⁺⁴, C₂².₈₉C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄×C₂³, C₂.C₂⁵, C₂².₈₉C₂⁵

Smallest permutation representation of C₂².₈₉C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2Q	4A	···	4L	4M	···	4Z
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

D₄

C₂.C₂⁵

kernel

C₂².₈₉C₂⁵

C₄×C₄○D₄

C₂×C₄.₄D₄

C₂².₂₆C₂⁴

C₂².₂₉C₂⁴

C₂³.₃₈C₂³

D₄⋊₅D₄

Q₈⋊₅D₄

C₂×2₊¹⁺⁴

C₂×2_-¹⁺⁴

C₄○D₄

C₂

# reps

Matrix representation of C₂².₈₉C₂⁵ ►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	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

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

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	2	0
0	0	0	0	0	2

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

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

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

C₂².₈₉C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{89}C_2^5

% in TeX

G:=Group("C2^2.89C2^5");

// GroupNames label

G:=SmallGroup(128,2232);

// by ID

G=gap.SmallGroup(128,2232);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,520,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b*a=a*b,g^2=a,d*c*d=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;

// generators/relations

G = C22.89C25 order 128 = 27

70th central stem extension by C22 of C25

G = C₂².₈₉C₂⁵ order 128 = 2⁷

70^th central stem extension by C₂² of C₂⁵