C2^3.372C2^4 - GroupNames

G = C₂³.₃₇₂C₂⁴ order 128 = 2⁷

89^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁵.₄₃C₂², C₂⁴.₅₇₁C₂³, C₂³.₃₇₂C₂⁴, C₂².₁₇₇2₊¹⁺⁴, C₂.₂₆D₄², C₂²⋊C₄⋊₄₂D₄, C₂⁴⋊₃C₄⋊₁₇C₂, (C₂×C₄²)⋊₂₄C₂², C₂³⋊Q₈⋊₁₃C₂, C₂³.₁₇₇(C₂×D₄), C₂.₅₄(D₄⋊₅D₄), (C₂²×Q₈)⋊₄C₂², C₂²⋊₂(C₄.₄D₄), (C₂³×C₄).₉₃C₂², C₂³.₂₃₅(C₄○D₄), C₂³.₂₃D₄⋊₄₉C₂, C₂³.₁₀D₄⋊₃₃C₂, C₂³.₁₁D₄⋊₁₉C₂, (C₂²×C₄).₅₁₆C₂³, C₂².₂₅₂(C₂²×D₄), C₂⁴.₃C₂²⋊₄₅C₂, C₂⁴.C₂²⋊₅₄C₂, C₂.C₄²⋊₆₇C₂², (C₂²×D₄).₁₃₉C₂², C₂.₁₇(C₂².₃₂C₂⁴), C₂.₄₄(C₂².₁₉C₂⁴), C₂.₂₁(C₂².₄₅C₂⁴), (C₂×C₄).₅₇(C₂×D₄), (C₂×C₄⋊C₄)⋊₁₈C₂², (C₄×C₂²⋊C₄)⋊₆₉C₂, (C₂×C₂²⋊Q₈)⋊₁₆C₂, (C₂×C₄.₄D₄)⋊₁₂C₂, C₂.₁₃(C₂×C₄.₄D₄), (C₂×C₂²≀C₂).₁₀C₂, (C₂²×C₂²⋊C₄)⋊₂₂C₂, (C₂×C₂²⋊C₄)⋊₁₉C₂², C₂².₂₄₉(C₂×C₄○D₄), SmallGroup(128,1204)

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

Derived series

C₁ — C₂³ — C₂³.₃₇₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₃₇₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₇₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₇₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₇₂C₂⁴

Generators and relations for C₂³.₃₇₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=b, f²=a, ab=ba, 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: 884 in 393 conjugacy classes, 112 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂⁵, C₄×C₂²⋊C₄, C₂⁴⋊₃C₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂²×C₂²⋊C₄, C₂×C₂²≀C₂, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂³.₃₇₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂×C₄.₄D₄, C₂².₃₂C₂⁴, D₄², D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂³.₃₇₂C₂⁴

Smallest permutation representation of C₂³.₃₇₂C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₃₇₂C₂⁴

C₄×C₂²⋊C₄

C₂⁴⋊₃C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂²×C₂²⋊C₄

C₂×C₂²≀C₂

C₂×C₂²⋊Q₈

C₂×C₄.₄D₄

C₂²⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₇₂C₂⁴ ►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	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

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

C₂³.₃₇₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{372}C_2^4

% in TeX

G:=Group("C2^3.372C2^4");

// GroupNames label

G:=SmallGroup(128,1204);

// by ID

G=gap.SmallGroup(128,1204);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=a,a*b=b*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 = C23.372C24 order 128 = 27

89th central stem extension by C23 of C24

G = C₂³.₃₇₂C₂⁴ order 128 = 2⁷

89^th central stem extension by C₂³ of C₂⁴