C2^2.81C2^5 - GroupNames

G = C₂².₈₁C₂⁵ order 128 = 2⁷

62^nd central stem extension by C₂² of C₂⁵

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, C₄²⋊C₂⋊₃₆C₂², C₂².₁₉C₂⁴⋊₂₇C₂, C₂²≀C₂.₃₆C₂², (C₂²×C₄).₃₅₄C₂³, (C₂³×C₄).₆₀₇C₂², C₂².D₄⋊₅C₂², C₂.₂₉(C₂×2₊¹⁺⁴), C₂.₂₀(C₂×2_-¹⁺⁴), C₂².₃₃C₂⁴⋊₃C₂, C₂³.₄₁C₂³⋊₁₁C₂, C₂².₃₁C₂⁴⋊₁₃C₂, C₂³.₃₃C₂³⋊₁₇C₂, C₂².₄₆C₂⁴⋊₁₃C₂, C₂².₄₇C₂⁴⋊₁₂C₂, C₄⋊C₄ ○C₂²≀C₂, (C₂×C₄)⋊₁₁(C₄○D₄), (C₂²×C₄⋊C₄)⋊₄₇C₂, (C₂×C₄⋊C₄)⋊₇₀C₂², C₄.₁₃₇(C₂×C₄○D₄), C₂.₄₆(C₂²×C₄○D₄), C₂².₁₄(C₂×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

Generators and relations for C₂².₈₁C₂⁵
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 >

Subgroups: 812 in 562 conjugacy classes, 392 normal (18 characteristic)
C₁, C₂, C₂, C₄, 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×Q₈, C₄○D₄, C₂⁴, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂³×C₄, C₂³×C₄, C₂×C₄○D₄, C₂²×C₄⋊C₄, C₂³.₃₃C₂³, C₂².₁₉C₂⁴, C₂².₃₁C₂⁴, C₂².₃₃C₂⁴, C₂³.₄₁C₂³, D₄⋊₆D₄, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, D₄⋊₃Q₈, C₂².₈₁C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂×2_-¹⁺⁴, C₂².₈₁C₂⁵

Smallest permutation representation of C₂².₈₁C₂⁵
►On 32 points

Generators in S₃₂

(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(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 21)(6 11)(7 23)(8 9)(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 8)(2 5 16 18)(3 19 13 6)(4 7 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)(5,18)(6,19)(7,20)(8,17)(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)(5,21)(6,11)(7,23)(8,9)(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), (1,17,15,8)(2,5,16,18)(3,19,13,6)(4,7,14,20)(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)(5,18)(6,19)(7,20)(8,17)(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)(5,21)(6,11)(7,23)(8,9)(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), (1,17,15,8)(2,5,16,18)(3,19,13,6)(4,7,14,20)(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),(5,18),(6,19),(7,20),(8,17),(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),(5,21),(6,11),(7,23),(8,9),(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)], [(1,17,15,8),(2,5,16,18),(3,19,13,6),(4,7,14,20),(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)]])

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

Matrix representation of C₂².₈₁C₂⁵ ►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	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] >;

C₂².₈₁C₂⁵ 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

62nd central stem extension by C22 of C25

G = C₂².₈₁C₂⁵ order 128 = 2⁷

62^nd central stem extension by C₂² of C₂⁵