C2^3.597C2^4 - GroupNames

G = C₂³.₅₉₇C₂⁴ order 128 = 2⁷

314^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₆₀C₂², C₂³.₅₉₇C₂⁴, C₂⁴.₄₀₄C₂³, C₂².₃₇₁2₊¹⁺⁴, (C₂×D₄).₁₄₀D₄, C₂⁴⋊₃C₄⋊₂₆C₂, (C₂³×C₄)⋊₁₃C₂², (C₂×C₄²)⋊₃₃C₂², C₂³.₂₁₂(C₂×D₄), C₂.₁₀₂(D₄⋊₅D₄), C₂³.Q₈⋊₆₁C₂, C₂³.₁₇₅(C₄○D₄), C₂³.₂₃D₄⋊₈₉C₂, C₂³.₁₁D₄⋊₈₆C₂, C₂³.₁₀D₄⋊₈₅C₂, C₂.₄₅(C₂³⋊₃D₄), (C₂²×C₄).₁₈₃C₂³, C₂².₄₀₆(C₂²×D₄), C₂.C₄²⋊₃₉C₂², C₂⁴.₃C₂²⋊₈₀C₂, (C₂²×D₄).₂₃₄C₂², C₂⁴.C₂²⋊₁₂₉C₂, C₂.₆₇(C₂².₃₂C₂⁴), C₂.₁₅(C₂².₅₄C₂⁴), C₂.₇₉(C₂².₄₅C₂⁴), (C₂×C₄).₉₉(C₂×D₄), (C₂×C₄⋊C₄)⋊₃₅C₂², (C₂×C₂²≀C₂).₁₅C₂, (C₂×C₂²⋊C₄)⋊₃₁C₂², C₂².₄₅₉(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₆C₂, SmallGroup(128,1429)

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

Derived series

C₁ — C₂³ — C₂³.₅₉₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴⋊₃C₄ — C₂³.₅₉₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₉₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₉₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₉₇C₂⁴

Generators and relations for C₂³.₅₉₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=e²=ba=ab, ac=ca, ede^-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 820 in 340 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂³.₅₉₇C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂².₅₄C₂⁴, C₂³.₅₉₇C₂⁴

Smallest permutation representation of C₂³.₅₉₇C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

(1 29)(2 17)(3 31)(4 19)(6 24)(8 22)(9 18)(10 32)(11 20)(12 30)(13 25)(15 27)

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4J	4K	···	4P
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₉₇C₂⁴

C₂⁴⋊₃C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₂²≀C₂

C₂×C₂².D₄

C₂×D₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₉₇C₂⁴ ►in GL₆(𝔽₅)

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
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	1

1	0	0	0	0	0
0	4	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

G:=sub<GL(6,GF(5))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C₂³.₅₉₇C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{597}C_2^4

% in TeX

G:=Group("C2^3.597C2^4");

// GroupNames label

G:=SmallGroup(128,1429);

// by ID

G=gap.SmallGroup(128,1429);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,1571,346]);

// 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=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*g=g*f>;

// generators/relations

G = C23.597C24 order 128 = 27

314th central stem extension by C23 of C24

G = C₂³.₅₉₇C₂⁴ order 128 = 2⁷

314^th central stem extension by C₂³ of C₂⁴