C2^2.97C2^5 - GroupNames

G = C₂².₉₇C₂⁵ order 128 = 2⁷

78^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₈₉C₂³, C₂².₉₇C₂⁵, C₂³.₄₇C₂⁴, C₂⁴.₁₃₈C₂³, C₄.₁₅₆2₊¹⁺⁴, D₄⋊₅D₄⋊₂₁C₂, Q₈⋊₆D₄⋊₂₀C₂, (C₄×D₄)⋊₄₈C₂², (C₂×C₄).₈₇C₂⁴, C₄⋊Q₈⋊₁₀₉C₂², (C₄×Q₈)⋊₄₇C₂², C₄⋊D₄⋊₈₄C₂², C₄⋊₁D₄⋊₁₈C₂², C₄⋊C₄.₄₉₂C₂³, (C₂×C₄²)⋊₆₂C₂², C₂²≀C₂.₉C₂², (C₂×D₄).₃₀₅C₂³, C₄.₄D₄⋊₈₅C₂², (C₂×Q₈).₄₅₂C₂³, C₄².C₂⋊₅₇C₂², C₂².₂₉C₂⁴⋊₂₅C₂, C₂².₁₁C₂⁴⋊₁₉C₂, C₂²⋊C₄.₁₀₇C₂³, (C₂²×C₄).₃₆₈C₂³, C₂².D₄⋊₉C₂², C₄²⋊C₂⋊₁₀₂C₂², C₂²⋊Q₈.₂₂₈C₂², C₂.₃₇(C₂×2₊¹⁺⁴), C₂².₂₆C₂⁴⋊₄₀C₂, (C₂²×D₄).₄₂₇C₂², C₂².₃₄C₂⁴⋊₁₀C₂, C₂³.₃₇C₂³⋊₄₀C₂, C₂².₅₃C₂⁴⋊₁₃C₂, (C₄×C₄○D₄)⋊₂₉C₂, (C₂×C₄)⋊₇(C₄○D₄), (C₂×C₄⋊₁D₄)⋊₂₇C₂, C₂²⋊C₄ ○(C₄⋊₁D₄), C₄.₁₈₀(C₂×C₄○D₄), (C₂×C₄○D₄)⋊₃₄C₂², C₂.₅₃(C₂²×C₄○D₄), C₂².₃₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₈₃C₂², SmallGroup(128,2240)

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

Derived series

C₁ — C₂² — C₂².₉₇C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₉₇C₂⁵

Lower central

C₁ — C₂² — C₂².₉₇C₂⁵

Upper central

C₁ — C₂² — C₂².₉₇C₂⁵

Jennings

C₁ — C₂² — C₂².₉₇C₂⁵

Generators and relations for C₂².₉₇C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=b, g²=a, ab=ba, 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: 1052 in 616 conjugacy classes, 392 normal (18 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, 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, C₄⋊₁D₄, C₄⋊Q₈, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, C₄×C₄○D₄, C₂².₁₁C₂⁴, C₂×C₄⋊₁D₄, C₂².₂₆C₂⁴, C₂³.₃₇C₂³, C₂².₂₉C₂⁴, C₂².₃₄C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₅₃C₂⁴, C₂².₉₇C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂².₉₇C₂⁵

Smallest permutation representation of C₂².₉₇C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

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

kernel

C₂².₉₇C₂⁵

C₄×C₄○D₄

C₂².₁₁C₂⁴

C₂×C₄⋊₁D₄

C₂².₂₆C₂⁴

C₂³.₃₇C₂³

C₂².₂₉C₂⁴

C₂².₃₄C₂⁴

D₄⋊₅D₄

Q₈⋊₆D₄

C₂².₅₃C₂⁴

C₂×C₄

C₄

# reps

Matrix representation of C₂².₉₇C₂⁵ ►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	4	0	0
0	0	4	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

3	0	0	0	0	0
0	3	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

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

C₂².₉₇C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{97}C_2^5

% in TeX

G:=Group("C2^2.97C2^5");

// GroupNames label

G:=SmallGroup(128,2240);

// by ID

G=gap.SmallGroup(128,2240);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b,g^2=a,a*b=b*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 = C22.97C25 order 128 = 27

78th central stem extension by C22 of C25

G = C₂².₉₇C₂⁵ order 128 = 2⁷

78^th central stem extension by C₂² of C₂⁵