C2^2.81C2^5

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

Aliases: C₄².₈₁C₂³, C₂².₈₁C₂⁵, C₂⁴.₆₂₀C₂³, C₂³.₁₃₂C₂⁴, C₂².₁2_-⁽¹⁺⁴⁾, C₂².₈2₊⁽¹⁺⁴⁾, C₄⋊Q₈⋊₃₁C₂², D₄⋊₆D₄⋊₁₉C₂, D₄⋊₃Q₈⋊₁₈C₂, (C₄×D₄)⋊₃₉C₂², (C₂×C₄).₇₃C₂⁴, (C₄×Q₈)⋊₄₀C₂², C₄⋊D₄⋊₂₃C₂², C₄⋊C₄.₄₈₅C₂³, C₂²⋊Q₈⋊₂₇C₂², C₄²⋊₂C₂⋊₃C₂², (C₂×D₄).₄₆₇C₂³, C₂²⋊C₄.₉₇C₂³, (C₂×Q₈).₂₈₄C₂³, C₄².C₂⋊₁₃C₂², C₂².₁₉C₂⁴⋊₂₇C₂, C₄²⋊C₂⋊₃₆C₂², C₂²≀C₂.₃₆C₂², (C₂²×C₄).₃₅₄C₂³, (C₂³×C₄).₆₀₇C₂², C₂².D₄⋊₅C₂², C₂.₂₀(C₂×2_-⁽¹⁺⁴⁾), C₂.₂₉(C₂×2₊⁽¹⁺⁴⁾), C₂².₃₃C₂⁴⋊₃C₂, C₂².₃₁C₂⁴⋊₁₃C₂, C₂².₄₇C₂⁴⋊₁₂C₂, C₂³.₄₁C₂³⋊₁₁C₂, C₂².₄₆C₂⁴⋊₁₃C₂, C₂³.₃₃C₂³⋊₁₇C₂, C₄⋊C₄ ○C₂²≀C₂, (C₂×C₄)⋊₁₁(C₄○D₄), (C₂×C₄⋊C₄)⋊₇₀C₂², (C₂²×C₄⋊C₄)⋊₄₇C₂, C₄.₁₃₇(C₂×C₄○D₄), C₂².₁₄(C₂×C₄○D₄), C₂.₄₆(C₂²×C₄○D₄), (C₂×C₄○D₄).₂₂₄C₂², SmallGroup(128,2224)

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

Derived series

C₁ — C₂² — C₂².₈₁C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂².₈₁C₂⁵

Lower central

C₁ — C₂² — C₂².₈₁C₂⁵

Upper central

C₁ — C₂² — C₂².₈₁C₂⁵

Jennings

C₁ — C₂² — C₂².₈₁C₂⁵

Subgroups: 812 in 562 conjugacy classes, 392 normal (18 characteristic)
C₁, C₂^[×3], C₂^[×10], C₄^[×4], C₄^[×22], C₂², C₂²^[×6], C₂²^[×26], C₂×C₄^[×28], C₂×C₄^[×42], D₄^[×32], Q₈^[×8], C₂³^[×3], C₂³^[×4], C₂³^[×6], C₄²^[×12], C₂²⋊C₄^[×36], C₄⋊C₄^[×52], C₂²×C₄^[×2], C₂²×C₄^[×28], C₂²×C₄^[×8], C₂×D₄^[×18], C₂×Q₈^[×6], C₄○D₄^[×16], C₂⁴, C₂×C₄⋊C₄^[×2], C₂×C₄⋊C₄^[×14], C₄²⋊C₂^[×10], C₄×D₄^[×28], C₄×Q₈^[×4], C₂²≀C₂^[×4], C₄⋊D₄^[×16], C₂²⋊Q₈^[×24], C₂².D₄^[×20], C₄².C₂^[×12], C₄²⋊₂C₂^[×8], C₄⋊Q₈^[×4], C₂³×C₄, C₂³×C₄^[×2], C₂×C₄○D₄^[×6], C₂²×C₄⋊C₄, C₂³.₃₃C₂³^[×2], C₂².₁₉C₂⁴^[×6], C₂².₃₁C₂⁴, C₂².₃₃C₂⁴^[×4], C₂³.₄₁C₂³, D₄⋊₆D₄^[×4], C₂².₄₆C₂⁴^[×4], C₂².₄₇C₂⁴^[×4], D₄⋊₃Q₈^[×4], C₂².₈₁C₂⁵

Quotients:
C₁, C₂^[×31], C₂²^[×155], C₂³^[×155], C₄○D₄^[×4], C₂⁴^[×31], C₂×C₄○D₄^[×6], 2₊⁽¹⁺⁴⁾^[×2], 2_-⁽¹⁺⁴⁾^[×2], C₂⁵, C₂²×C₄○D₄, C₂×2₊⁽¹⁺⁴⁾, C₂×2_-⁽¹⁺⁴⁾, C₂².₈₁C₂⁵

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

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

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

(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 3)(2 4)(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)

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

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

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

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

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

Matrix representation ►G ⊆ 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	1	0	0	0	0
1	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

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

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
0	0	0	0	0	1

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

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

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

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂².₈₁C₂⁵

C₂²×C₄⋊C₄

C₂³.₃₃C₂³

C₂².₁₉C₂⁴

C₂².₃₁C₂⁴

C₂².₃₃C₂⁴

C₂³.₄₁C₂³

D₄⋊₆D₄

C₂².₄₆C₂⁴

C₂².₄₇C₂⁴

D₄⋊₃Q₈

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^2._{81}C_2^5

% in TeX

G:=Group("C2^2.81C2^5");

// GroupNames label

G:=SmallGroup(128,2224);

// by ID

G=gap.SmallGroup(128,2224);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,136,1684]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=g^2=1,d^2=b,f^2=a,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=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 = C₂².₈₁C₂⁵ order 128 = 2⁷

62^nd central stem extension by C₂² of C₂⁵

G = C22.81C25 order 128 = 27

62nd central stem extension by C22 of C25

G = C₂².₈₁C₂⁵ order 128 = 2⁷

62^nd central stem extension by C₂² of C₂⁵