C2^2.90C2^5 - GroupNames

G = C₂².₉₀C₂⁵ order 128 = 2⁷

71^st central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₉₀C₂⁵, C₂³.₁₃₄C₂⁴, C₄².₅₇₈C₂³, C₂⁴.₅₀₇C₂³, C₄.₇₉2₊¹⁺⁴, D₄⋊₈(C₂×Q₈), (C₂×D₄)⋊₂₃Q₈, (D₄×Q₈)⋊₁₉C₂, C₂³⋊₂(C₂×Q₈), D₄ ○₂(C₂²⋊Q₈), C₄⋊Q₈⋊₉₂C₂², D₄⋊₃Q₈⋊₂₀C₂, (C₂×C₄).₈₀C₂⁴, (C₄×Q₈)⋊₄₄C₂², C₂³⋊₂Q₈⋊₆C₂, C₂.₁₅(Q₈×C₂³), C₄.₅₃(C₂²×Q₈), C₄⋊C₄.₂₉₆C₂³, C₂²⋊Q₈⋊₃₃C₂², (C₂×D₄).₅₀₅C₂³, (C₄×D₄).₂₃₂C₂², (C₂×Q₈).₂₈₇C₂³, C₄².C₂⋊₁₅C₂², (C₂²×Q₈)⋊₃₃C₂², C₂².₁₁(C₂²×Q₈), C₂²⋊C₄.₁₀₀C₂³, (C₂²×C₄).₃₆₂C₂³, (C₂³×C₄).₆₁₁C₂², (C₂×C₄²).₉₄₄C₂², C₂.₃₃(C₂×2₊¹⁺⁴), C₂.₂₅(C₂.C₂⁵), C₂².₁₁C₂⁴.₁₀C₂, (C₂²×D₄).₅₉₉C₂², C₂³.₃₇C₂³⋊₃₅C₂, C₄²⋊C₂.₂₂₅C₂², C₂³.₄₁C₂³⋊₁₅C₂, (C₂×C₄)⋊₃(C₂×Q₈), (C₂×C₄×D₄).₉₀C₂, (C₂×D₄)○(C₂²⋊Q₈), (C₂×C₄⋊C₄)⋊₇₄C₂², (C₂×C₂²⋊Q₈)⋊₇₇C₂, (C₂×C₂²⋊C₄).₃₈₂C₂², SmallGroup(128,2233)

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

Derived series

C₁ — C₂² — C₂².₉₀C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄×D₄ — C₂².₉₀C₂⁵

Lower central

C₁ — C₂² — C₂².₉₀C₂⁵

Upper central

C₁ — C₂² — C₂².₉₀C₂⁵

Jennings

C₁ — C₂² — C₂².₉₀C₂⁵

Generators and relations for C₂².₉₀C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=d²=f²=g²=1, c²=e²=b, ab=ba, dcd=gcg=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: 788 in 550 conjugacy classes, 430 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄×D₄, C₂².₁₁C₂⁴, C₂×C₂²⋊Q₈, C₂³.₃₇C₂³, C₂³⋊₂Q₈, C₂³.₄₁C₂³, D₄×Q₈, D₄⋊₃Q₈, C₂².₉₀C₂⁵
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₂⁴, C₂²×Q₈, 2₊¹⁺⁴, C₂⁵, Q₈×C₂³, C₂×2₊¹⁺⁴, C₂.C₂⁵, C₂².₉₀C₂⁵

Smallest permutation representation of C₂².₉₀C₂⁵
►On 32 points

Generators in S₃₂

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

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

(5 20)(6 17)(7 18)(8 19)(21 29)(22 30)(23 31)(24 32)

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	4A	···	4H	4I	···	4AD
order	1	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

Q₈

2₊¹⁺⁴

C₂.C₂⁵

kernel

C₂².₉₀C₂⁵

C₂×C₄×D₄

C₂².₁₁C₂⁴

C₂×C₂²⋊Q₈

C₂³.₃₇C₂³

C₂³⋊₂Q₈

C₂³.₄₁C₂³

D₄×Q₈

D₄⋊₃Q₈

C₂×D₄

C₄

C₂

# reps

Matrix representation of C₂².₉₀C₂⁵ ►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

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

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

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

C₂².₉₀C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{90}C_2^5

% in TeX

G:=Group("C2^2.90C2^5");

// GroupNames label

G:=SmallGroup(128,2233);

// by ID

G=gap.SmallGroup(128,2233);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,352,570,1684]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=g^2=1,c^2=e^2=b,a*b=b*a,d*c*d=g*c*g=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 = C22.90C25 order 128 = 27

71st central stem extension by C22 of C25

G = C₂².₉₀C₂⁵ order 128 = 2⁷

71^st central stem extension by C₂² of C₂⁵