C2^2.94C2^5 - GroupNames

G = C₂².₉₄C₂⁵ order 128 = 2⁷

75^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₉₄C₂⁵, C₄².₈₆C₂³, C₂⁴.₁₃₇C₂³, C₂³.₁₃₈C₂⁴, C₄.₈₀2₊¹⁺⁴, D₄⋊₅D₄⋊₂₀C₂, Q₈⋊₅D₄⋊₁₈C₂, (C₄×D₄)⋊₄₆C₂², C₂³⋊₄(C₄○D₄), C₂³⋊₃D₄⋊₈C₂, (C₂×C₄).₈₄C₂⁴, (C₄×Q₈)⋊₄₅C₂², C₄⋊C₄.₄₉₀C₂³, C₄⋊D₄⋊₈₂C₂², (C₂³×C₄)⋊₄₃C₂², (C₂×C₄²)⋊₆₁C₂², C₂²⋊Q₈⋊₃₄C₂², C₂²≀C₂⋊₃₄C₂², C₂².₃₂C₂⁴⋊₄C₂, C₄²⋊₂C₂⋊₄C₂², (C₂×D₄).₃₀₃C₂³, C₄.₄D₄⋊₈₄C₂², (C₂²×D₄)⋊₃₈C₂², (C₂×Q₈).₄₅₀C₂³, C₄².C₂⋊₅₅C₂², (C₂²×Q₈)⋊₃₄C₂², C₂².₄₅C₂⁴⋊₆C₂, C₂².₁₉C₂⁴⋊₃₀C₂, C₄²⋊C₂⋊₄₀C₂², C₂².₁₁C₂⁴⋊₁₈C₂, C₂²⋊C₄.₁₀₄C₂³, (C₂²×C₄).₃₆₅C₂³, C₂².D₄⋊₈C₂², C₂.₃₅(C₂×2₊¹⁺⁴), C₂.₂₈(C₂.C₂⁵), C₂².₃₃C₂⁴⋊₄C₂, C₂³.₃₆C₂³⋊₂₉C₂, C₂².₄₇C₂⁴⋊₁₅C₂, C₂².₄₆C₂⁴⋊₁₆C₂, (C₂×C₄×D₄)⋊₉₁C₂, (C₂×C₄⋊D₄)⋊₆₈C₂, (C₂×C₄⋊C₄)⋊₇₅C₂², (C₂×C₂²⋊Q₈)⋊₇₈C₂, C₂²⋊C₄ ○(C₂²⋊Q₈), (C₂×C₄○D₄)⋊₃₂C₂², C₂.₅₀(C₂²×C₄○D₄), C₂².₃₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄)⋊₅₀C₂², SmallGroup(128,2237)

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

Derived series

C₁ — C₂² — C₂².₉₄C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄×D₄ — C₂².₉₄C₂⁵

Lower central

C₁ — C₂² — C₂².₉₄C₂⁵

Upper central

C₁ — C₂² — C₂².₉₄C₂⁵

Jennings

C₁ — C₂² — C₂².₉₄C₂⁵

Generators and relations for C₂².₉₄C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=e²=b, 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: 916 in 576 conjugacy classes, 390 normal (38 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×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, 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, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄×D₄, C₂².₁₁C₂⁴, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂³⋊₃D₄, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₉₄C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂.C₂⁵, C₂².₉₄C₂⁵

Smallest permutation representation of C₂².₉₄C₂⁵
►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 23)(2 32)(3 21)(4 30)(5 10)(6 15)(7 12)(8 13)(9 19)(11 17)(14 20)(16 18)(22 26)(24 28)(25 29)(27 31)

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

(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)

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

C₂².₉₄C₂⁵

C₂×C₄×D₄

C₂².₁₁C₂⁴

C₂×C₄⋊D₄

C₂×C₂²⋊Q₈

C₂².₁₉C₂⁴

C₂³.₃₆C₂³

C₂³⋊₃D₄

C₂².₃₂C₂⁴

C₂².₃₃C₂⁴

D₄⋊₅D₄

Q₈⋊₅D₄

C₂².₄₅C₂⁴

C₂².₄₆C₂⁴

C₂².₄₇C₂⁴

C₂³

C₄

C₂

# reps

Matrix representation of C₂².₉₄C₂⁵ ►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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂².₉₄C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{94}C_2^5

% in TeX

G:=Group("C2^2.94C2^5");

// GroupNames label

G:=SmallGroup(128,2237);

// by ID

G=gap.SmallGroup(128,2237);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=b,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 = C22.94C25 order 128 = 27

75th central stem extension by C22 of C25

G = C₂².₉₄C₂⁵ order 128 = 2⁷

75^th central stem extension by C₂² of C₂⁵