C2^3.382C2^4 - GroupNames

G = C₂³.₃₈₂C₂⁴ order 128 = 2⁷

99^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₄₅C₂², C₂³.₃₈₂C₂⁴, C₂⁴.₅₇₄C₂³, C₂².₁₈₅2₊¹⁺⁴, (C₂×C₄²)⋊₆C₂², C₂⁴⋊₃C₄.₉C₂, (C₂²×C₄).₃₈₀D₄, C₂³.₆₁₁(C₂×D₄), (C₂²×C₄).₆₈C₂³, C₂³.₄Q₈⋊₁₃C₂, C₂³.₈Q₈⋊₆₁C₂, C₂³.₁₄₅(C₄○D₄), C₂³.₁₁D₄⋊₂₄C₂, C₂³.₃₄D₄⋊₂₉C₂, (C₂³×C₄).₃₆₉C₂², C₂².₂₆₂(C₂²×D₄), C₂⁴.C₂²⋊₆₁C₂, C₂.C₄²⋊₂₄C₂², C₂.₅₄(C₂².₁₉C₂⁴), C₂.₁₅(C₂².₂₉C₂⁴), C₂.₂₅(C₂².₄₅C₂⁴), C₂².₆₁(C₂².D₄), (C₂×C₄).₃₄₇(C₂×D₄), (C₂×C₄⋊C₄)⋊₁₁₁C₂², (C₂×C₄²⋊C₂)⋊₂₈C₂, C₂².₂₅₉(C₂×C₄○D₄), (C₂²×C₂²⋊C₄).₂₄C₂, C₂.₂₇(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₁₅₀C₂², SmallGroup(128,1214)

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

Derived series

C₁ — C₂³ — C₂³.₃₈₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³.₃₈₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₈₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₈₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₈₂C₂⁴

Generators and relations for C₂³.₃₈₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=g²=1, d²=c, e²=a, ab=ba, ac=ca, ede^-1=gdg=ad=da, 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, eg=ge, fg=gf >

Subgroups: 724 in 336 conjugacy classes, 104 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂³×C₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂⁴.C₂², C₂³.₁₁D₄, C₂³.₄Q₈, C₂²×C₂²⋊C₄, C₂×C₄²⋊C₂, C₂³.₃₈₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂².₂₉C₂⁴, C₂².₄₅C₂⁴, C₂³.₃₈₂C₂⁴

Smallest permutation representation of C₂³.₃₈₂C₂⁴
►On 32 points

Generators in S₃₂

(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)

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

(1 27)(3 25)(5 7)(6 19)(8 17)(9 13)(11 15)(18 20)(21 31)(22 24)(23 29)(30 32)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	4E	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	4	4	4	4	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₈₂C₂⁴

C₂⁴⋊₃C₄

C₂³.₃₄D₄

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.₁₁D₄

C₂³.₄Q₈

C₂²×C₂²⋊C₄

C₂×C₄²⋊C₂

C₂²×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₈₂C₂⁴ ►in GL₆(𝔽₅)

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

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

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	1	0
0	0	0	0	0	1

0	3	0	0	0	0
3	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

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

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	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	4

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

C₂³.₃₈₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{382}C_2^4

% in TeX

G:=Group("C2^3.382C2^4");

// GroupNames label

G:=SmallGroup(128,1214);

// by ID

G=gap.SmallGroup(128,1214);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c,e^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,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,e*g=g*e,f*g=g*f>;

// generators/relations

G = C23.382C24 order 128 = 27

99th central stem extension by C23 of C24

G = C₂³.₃₈₂C₂⁴ order 128 = 2⁷

99^th central stem extension by C₂³ of C₂⁴