C2^2.99C2^5 - GroupNames

G = C₂².₉₉C₂⁵ order 128 = 2⁷

80^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₉₁C₂³, C₂².₉₉C₂⁵, C₂³.₁₄₀C₂⁴, C₂⁴.₁₃₉C₂³, C₄⋊Q₈⋊₃₄C₂², (C₄×D₄)⋊₄₉C₂², (C₂×C₄).₈₉C₂⁴, (C₄×Q₈)⋊₄₈C₂², C₄⋊C₄.₄₉₄C₂³, (C₂×C₄²)⋊₆₃C₂², C₂²⋊Q₈⋊₃₅C₂², (C₂×D₄).₃₀₆C₂³, C₄.₄D₄⋊₂₈C₂², (C₂×Q₈).₂₉₁C₂³, C₄².C₂⋊₇₅C₂², C₂².₄₅C₂⁴⋊₇C₂, C₄²⋊₂C₂⋊₃₇C₂², C₂².₁₁C₂⁴⋊₂₀C₂, C₄²⋊C₂⋊₄₂C₂², C₂²≀C₂.₁₀C₂², C₄⋊D₄.₂₂₆C₂², C₄⋊₁D₄.₁₁₃C₂², C₂²⋊C₄.₁₀₈C₂³, (C₂²×C₄).₃₆₉C₂³, C₂.₃₁(C₂.C₂⁵), C₂².₂₉C₂⁴.₁₆C₂, (C₂²×D₄).₄₂₈C₂², C₂².D₄⋊₅₃C₂², (C₂²×Q₈).₃₆₂C₂², C₂².₅₀C₂⁴⋊₂₃C₂, C₂³.₃₇C₂³⋊₄₂C₂, C₂³.₃₂C₂³⋊₁₅C₂, C₂².₃₆C₂⁴⋊₁₄C₂, C₂³.₃₆C₂³⋊₃₁C₂, C₂².₅₃C₂⁴⋊₁₄C₂, C₂³.₃₈C₂³⋊₂₅C₂, (C₄×C₄○D₄)⋊₃₀C₂, C₄.₁₈₂(C₂×C₄○D₄), (C₂×C₄.₄D₄)⋊₅₅C₂, C₂².₃₂(C₂×C₄○D₄), C₂.₅₅(C₂²×C₄○D₄), (C₂×C₄).₃₀₈(C₄○D₄), (C₂×C₄○D₄).₃₃₁C₂², (C₂×C₂²⋊C₄).₃₈₄C₂², SmallGroup(128,2242)

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

Derived series

C₁ — C₂² — C₂².₉₉C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₉₉C₂⁵

Lower central

C₁ — C₂² — C₂².₉₉C₂⁵

Upper central

C₁ — C₂² — C₂².₉₉C₂⁵

Jennings

C₁ — C₂² — C₂².₉₉C₂⁵

Generators and relations for C₂².₉₉C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=b, e²=ba=ab, g²=a, dcd^-1=gcg^-1=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: 748 in 520 conjugacy classes, 388 normal (26 characteristic)
C₁, C₂, 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, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₄×C₄○D₄, C₂².₁₁C₂⁴, C₂³.₃₂C₂³, C₂×C₄.₄D₄, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₂₉C₂⁴, C₂³.₃₈C₂³, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂².₉₉C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, C₂⁵, C₂²×C₄○D₄, C₂.C₂⁵, C₂².₉₉C₂⁵

Smallest permutation representation of C₂².₉₉C₂⁵
►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 11)(2 16)(3 9)(4 14)(5 32)(6 21)(7 30)(8 23)(10 26)(12 28)(13 25)(15 27)(17 29)(18 22)(19 31)(20 24)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2K	4A	···	4P	4Q	···	4AF
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

C₂.C₂⁵

kernel

C₂².₉₉C₂⁵

C₄×C₄○D₄

C₂².₁₁C₂⁴

C₂³.₃₂C₂³

C₂×C₄.₄D₄

C₂³.₃₆C₂³

C₂³.₃₇C₂³

C₂².₂₉C₂⁴

C₂³.₃₈C₂³

C₂².₃₆C₂⁴

C₂².₄₅C₂⁴

C₂².₅₀C₂⁴

C₂².₅₃C₂⁴

C₂×C₄

C₂

# reps

Matrix representation of C₂².₉₉C₂⁵ ►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

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

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

3	2	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	0
0	0	0	0	0	2

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	4	0	4	0
0	0	0	4	0	4

4	0	0	0	0	0
0	4	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

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

C₂².₉₉C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{99}C_2^5

% in TeX

G:=Group("C2^2.99C2^5");

// GroupNames label

G:=SmallGroup(128,2242);

// by ID

G=gap.SmallGroup(128,2242);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,520,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=b,e^2=b*a=a*b,g^2=a,d*c*d^-1=g*c*g^-1=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 = C22.99C25 order 128 = 27

80th central stem extension by C22 of C25

G = C₂².₉₉C₂⁵ order 128 = 2⁷

80^th central stem extension by C₂² of C₂⁵