C2^2.95C2^5 - GroupNames

G = C₂².₉₅C₂⁵ order 128 = 2⁷

76^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₈₇C₂³, C₂².₉₅C₂⁵, C₂³.₄₆C₂⁴, C₂⁴.₅₀₈C₂³, C₄.₈₁2₊¹⁺⁴, C₄⋊Q₈⋊₉₃C₂², D₄⋊₆D₄⋊₂₃C₂, Q₈⋊₆D₄⋊₁₉C₂, (C₄×D₄)⋊₄₇C₂², (C₂×C₄).₈₅C₂⁴, (C₄×Q₈)⋊₄₆C₂², C₄⋊₁D₄⋊₅₃C₂², C₄⋊D₄⋊₈₃C₂², C₄⋊C₄.₄₉₁C₂³, C₂²⋊Q₈⋊₉₆C₂², C₄²⋊₂C₂⋊₅C₂², C₂².₃₂C₂⁴⋊₅C₂, (C₂×D₄).₃₀₄C₂³, C₄.₄D₄⋊₂₇C₂², (C₂×Q₈).₂₉₀C₂³, C₄².C₂⋊₅₆C₂², C₄²⋊C₂⋊₄₁C₂², C₂².₁₉C₂⁴⋊₃₁C₂, C₂².₂₉C₂⁴⋊₂₄C₂, C₂²≀C₂.₂₉C₂², C₂²⋊C₄.₁₀₅C₂³, (C₂×C₄²).₉₄₈C₂², (C₂²×C₄).₃₆₆C₂³, (C₂³×C₄).₆₁₂C₂², C₂.₃₆(C₂×2₊¹⁺⁴), C₂.₂₉(C₂.C₂⁵), C₂².₂₆C₂⁴⋊₃₉C₂, (C₂²×D₄).₆₀₀C₂², C₂².D₄⋊₅₂C₂², C₂².₄₇C₂⁴⋊₁₆C₂, C₂².₅₀C₂⁴⋊₂₀C₂, C₂³.₃₇C₂³⋊₃₈C₂, C₂².₅₃C₂⁴⋊₁₁C₂, C₂³.₃₃C₂³⋊₂₁C₂, C₂².₄₉C₂⁴⋊₁₃C₂, (C₂×C₄×D₄)⋊₉₂C₂, (C₂×C₄)⋊₆(C₄○D₄), (C₂×C₄⋊C₄)⋊₇₆C₂², C₄.₁₇₈(C₂×C₄○D₄), (C₂×C₄○D₄)⋊₃₃C₂², C₂².₁₆(C₂×C₄○D₄), C₂.₅₁(C₂²×C₄○D₄), (C₂×C₂²⋊C₄).₅₄₇C₂², SmallGroup(128,2238)

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

Derived series

C₁ — C₂² — C₂².₉₅C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂×C₄×D₄ — C₂².₉₅C₂⁵

Lower central

C₁ — C₂² — C₂².₉₅C₂⁵

Upper central

C₁ — C₂² — C₂².₉₅C₂⁵

Jennings

C₁ — C₂² — C₂².₉₅C₂⁵

Generators and relations for C₂².₉₅C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=e²=b, f²=a, ab=ba, dcd^-1=gcg=ac=ca, fdf^-1=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: 884 in 570 conjugacy classes, 390 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄×D₄, C₂³.₃₃C₂³, C₂².₁₉C₂⁴, C₂².₂₆C₂⁴, C₂³.₃₇C₂³, C₂².₂₉C₂⁴, C₂².₃₂C₂⁴, D₄⋊₆D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂².₉₅C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂.C₂⁵, C₂².₉₅C₂⁵

Smallest permutation representation of C₂².₉₅C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2M	4A	···	4P	4Q	···	4AD
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

C₄○D₄

2₊¹⁺⁴

C₂.C₂⁵

kernel

C₂².₉₅C₂⁵

C₂×C₄×D₄

C₂³.₃₃C₂³

C₂².₁₉C₂⁴

C₂².₂₆C₂⁴

C₂³.₃₇C₂³

C₂².₂₉C₂⁴

C₂².₃₂C₂⁴

D₄⋊₆D₄

Q₈⋊₆D₄

C₂².₄₇C₂⁴

C₂².₄₉C₂⁴

C₂².₅₀C₂⁴

C₂².₅₃C₂⁴

C₂×C₄

C₄

C₂

# reps

Matrix representation of C₂².₉₅C₂⁵ ►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	4	0	0	0	0
4	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	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	4
0	0	0	0	4	0

3	0	0	0	0	0
0	2	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

1	0	0	0	0	0
0	1	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

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

C₂².₉₅C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{95}C_2^5

% in TeX

G:=Group("C2^2.95C2^5");

// GroupNames label

G:=SmallGroup(128,2238);

// by ID

G=gap.SmallGroup(128,2238);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,136,1684]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^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^-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 = C22.95C25 order 128 = 27

76th central stem extension by C22 of C25

G = C₂².₉₅C₂⁵ order 128 = 2⁷

76^th central stem extension by C₂² of C₂⁵