C2^2.80C2^5 - GroupNames

G = C₂².₈₀C₂⁵ order 128 = 2⁷

61^st central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₈₀C₂⁵, C₂³.₃₈C₂⁴, C₄².₅₇₀C₂³, C₂⁴.₁₃₃C₂³, (C₄×D₄)⋊₃₈C₂², (C₄×Q₈)⋊₃₉C₂², C₂³⋊₂Q₈⋊₅C₂, C₄⋊C₄.₄₈₄C₂³, (C₂×C₄).₁₇₀C₂⁴, (C₂×C₄²)⋊₅₆C₂², C₂³⋊₃D₄.₈C₂, C₂²⋊Q₈⋊₂₆C₂², C₄²⋊₂C₂⋊₂C₂², C₂²≀C₂.₇C₂², (C₂×D₄).₄₆₆C₂³, C₄.₄D₄⋊₇₈C₂², C₂²⋊C₄.₉₆C₂³, (C₂×Q₈).₂₈₃C₂³, C₄².C₂⋊₅₂C₂², C₄²⋊C₂⋊₃₅C₂², C₂².₄₅C₂⁴⋊₃C₂, C₂².₁₁C₂⁴⋊₁₆C₂, C₂³.₁₉₈(C₄○D₄), C₄⋊D₄.₂₂₄C₂², (C₂³×C₄).₆₀₆C₂², (C₂²×C₄).₃₅₃C₂³, C₂².D₄⋊₄C₂², C₂.₁₈(C₂.C₂⁵), C₂².₃₃C₂⁴⋊₂C₂, (C₂²×D₄).₄₂₄C₂², C₂².₄₆C₂⁴⋊₁₂C₂, C₂³.₃₆C₂³⋊₂₅C₂, (C₂×C₄⋊C₄)⋊₁₄₁C₂², C₂².₂₇(C₂×C₄○D₄), C₂.₄₅(C₂²×C₄○D₄), (C₂×C₄²⋊C₂)⋊₆₅C₂, (C₂×C₂²⋊C₄)⋊₄₇C₂², (C₂×C₂².D₄)⋊₆₀C₂, C₂²⋊C₄ ○(C₂².D₄), SmallGroup(128,2223)

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

Derived series

C₁ — C₂² — C₂².₈₀C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₂².₈₀C₂⁵

Lower central

C₁ — C₂² — C₂².₈₀C₂⁵

Upper central

C₁ — C₂² — C₂².₈₀C₂⁵

Jennings

C₁ — C₂² — C₂².₈₀C₂⁵

Generators and relations for C₂².₈₀C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=b, e²=a, ab=ba, dcd^-1=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: 764 in 526 conjugacy classes, 388 normal (14 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, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂×C₄²⋊C₂, C₂².₁₁C₂⁴, C₂×C₂².D₄, C₂³.₃₆C₂³, C₂³⋊₃D₄, C₂².₃₃C₂⁴, C₂³⋊₂Q₈, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₈₀C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, C₂⁵, C₂²×C₄○D₄, C₂.C₂⁵, C₂².₈₀C₂⁵

Smallest permutation representation of C₂².₈₀C₂⁵
►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 22)(2 31)(3 24)(4 29)(5 15)(6 12)(7 13)(8 10)(9 18)(11 20)(14 19)(16 17)(21 26)(23 28)(25 32)(27 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 27 23)(2 32 28 24)(3 29 25 21)(4 30 26 22)(5 10 20 14)(6 11 17 15)(7 12 18 16)(8 9 19 13)

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	···	4L	4M	···	4AD
order	1	2	2	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	4	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

C₂.C₂⁵

kernel

C₂².₈₀C₂⁵

C₂×C₄²⋊C₂

C₂².₁₁C₂⁴

C₂×C₂².D₄

C₂³.₃₆C₂³

C₂³⋊₃D₄

C₂².₃₃C₂⁴

C₂³⋊₂Q₈

C₂².₄₅C₂⁴

C₂².₄₆C₂⁴

C₂³

C₂

# reps

Matrix representation of C₂².₈₀C₂⁵ ►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
2	0	0	0	0	0
0	0	0	0	2	0
0	0	1	4	3	1
0	0	3	0	0	0
0	0	2	0	1	1

3	0	0	0	0	0
0	3	0	0	0	0
0	0	0	0	1	0
0	0	2	3	1	2
0	0	1	0	0	0
0	0	4	1	0	2

0	1	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

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	3	2	4	3
0	0	2	3	0	1

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

C₂².₈₀C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{80}C_2^5

% in TeX

G:=Group("C2^2.80C2^5");

// GroupNames label

G:=SmallGroup(128,2223);

// by ID

G=gap.SmallGroup(128,2223);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,1684]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=b,e^2=a,a*b=b*a,d*c*d^-1=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 = C22.80C25 order 128 = 27

61st central stem extension by C22 of C25

G = C₂².₈₀C₂⁵ order 128 = 2⁷

61^st central stem extension by C₂² of C₂⁵