C2^3.380C2^4 - GroupNames

G = C₂³.₃₈₀C₂⁴ order 128 = 2⁷

97^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₄₄C₂², C₂⁴.₂₉₇C₂³, C₂³.₃₈₀C₂⁴, C₂².₁₈₃2₊¹⁺⁴, (C₂×C₄²)⋊₅C₂², C₂⁴⋊₃C₄.₈C₂, C₂²⋊C₄.₁₃₅D₄, C₂³.₁₈₁(C₂×D₄), C₂.₅₉(D₄⋊₅D₄), (C₂²×C₄).₆₆C₂³, (C₂³×C₄).₉₄C₂², C₂³.₇Q₈⋊₅₁C₂, C₂³.Q₈⋊₂₁C₂, C₂³.₈Q₈⋊₅₉C₂, C₂²⋊₂(C₄²⋊₂C₂), C₂³.₁₄₄(C₄○D₄), C₂³.₁₁D₄⋊₂₂C₂, C₂².₂₆₀(C₂²×D₄), C₂.C₄²⋊₅₅C₂², C₂⁴.C₂²⋊₅₉C₂, C₂.₁₈(C₂².₃₂C₂⁴), C₂.₅₂(C₂².₁₉C₂⁴), C₂.₂₄(C₂².₄₅C₂⁴), (C₄×C₂²⋊C₄)⋊₁₃C₂, (C₂×C₄⋊C₄)⋊₁₉C₂², (C₂×C₄).₉₀₄(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₅C₂, C₂.₁₀(C₂×C₄²⋊₂C₂), C₂².₂₅₇(C₂×C₄○D₄), (C₂²×C₂²⋊C₄).₂₃C₂, (C₂×C₂²⋊C₄).₁₄₈C₂², SmallGroup(128,1212)

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

Derived series

C₁ — C₂³ — C₂³.₃₈₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₃₈₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₈₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₈₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₈₀C₂⁴

Generators and relations for C₂³.₃₈₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=ba=ab, f²=a, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 724 in 332 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂³×C₄, C₂⁵, C₄×C₂²⋊C₄, C₂⁴⋊₃C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂⁴.C₂², C₂³.Q₈, C₂³.₁₁D₄, C₂²×C₂²⋊C₄, C₂×C₄²⋊₂C₂, C₂³.₃₈₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂×C₄²⋊₂C₂, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂³.₃₈₀C₂⁴

Smallest permutation representation of C₂³.₃₈₀C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

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

C₂³.₃₈₀C₂⁴

C₄×C₂²⋊C₄

C₂⁴⋊₃C₄

C₂³.₇Q₈

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.Q₈

C₂³.₁₁D₄

C₂²×C₂²⋊C₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₈₀C₂⁴ ►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	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	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	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
4	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	4

4	3	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	0	1
0	0	0	0	1	0

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

4	3	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	0	1
0	0	0	0	1	0

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

C₂³.₃₈₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{380}C_2^4

% in TeX

G:=Group("C2^3.380C2^4");

// GroupNames label

G:=SmallGroup(128,1212);

// by ID

G=gap.SmallGroup(128,1212);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,675,192]);

// Polycyclic

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

// generators/relations

G = C23.380C24 order 128 = 27

97th central stem extension by C23 of C24

G = C₂³.₃₈₀C₂⁴ order 128 = 2⁷

97^th central stem extension by C₂³ of C₂⁴