C2^4.174C2^3 - GroupNames

G = C₂⁴.₁₇₄C₂³ order 128 = 2⁷

14^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂⁴.₁₇₄C₂³, C₂⁴.₇(C₂×C₄), C₂².₅₀(C₄×D₄), C₂²⋊C₄.₁₁₉D₄, C₂³.₉D₄⋊₉C₂, (C₂²×C₄).₂₉₃D₄, C₂³.₁₂₈(C₂×D₄), C₂².₃₀C₂²≀C₂, C₂³.₃₄D₄⋊₇C₂, C₂³.₁₂₁(C₄○D₄), C₂².₄₈(C₄⋊D₄), C₂³.₁₉₃(C₂²×C₄), (C₂³×C₄).₂₆₂C₂², (C₂²×D₄).₂₇C₂², C₂.₃₈(C₂³.₂₃D₄), C₂.₂₇(C₂³.C₂³), C₂².₂₈(C₂².D₄), (C₂×C₄⋊C₄)⋊₁₃C₄, (C₂×C₂²⋊C₄)⋊₅C₄, (C₂×C₂³⋊C₄).₇C₂, (C₂×C₄²⋊C₂)⋊₃C₂, (C₂²×C₄).₁₉(C₂×C₄), (C₂×C₄).₁₉₆(C₂²⋊C₄), (C₂×C₂²⋊C₄).₁₃C₂², C₂².₂₇₄(C₂×C₂²⋊C₄), (C₂×C₂².D₄).₃C₂, SmallGroup(128,631)

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

Derived series

C₁ — C₂³ — C₂⁴.₁₇₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₂⁴.₁₇₄C₂³

Lower central

C₁ — C₂ — C₂³ — C₂⁴.₁₇₄C₂³

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₇₄C₂³

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₁₇₄C₂³

Generators and relations for C₂⁴.₁₇₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=ca=ac, g²=d, ab=ba, eae^-1=faf^-1=ad=da, ag=ga, bc=cb, bd=db, geg^-1=be=eb, bf=fb, bg=gb, fcf^-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef^-1=ae, fg=gf >

Subgroups: 420 in 189 conjugacy classes, 56 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₉D₄, C₂³.₃₄D₄, C₂×C₂³⋊C₄, C₂×C₄²⋊C₂, C₂×C₂².D₄, C₂⁴.₁₇₄C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³.₂₃D₄, C₂³.C₂³, C₂⁴.₁₇₄C₂³

Smallest permutation representation of C₂⁴.₁₇₄C₂³
►On 32 points

Generators in S₃₂

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

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

(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 12)(10 11)(13 16)(14 15)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	4A	4B	4C	4D	4E	···	4N	4O	···	4U
order	1	2	2	2	2	···	2	2	4	4	4	4	4	···	4	4	···	4
size	1	1	1	1	2	···	2	8	2	2	2	2	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

C₄○D₄

C₂³.C₂³

kernel

C₂⁴.₁₇₄C₂³

C₂³.₉D₄

C₂³.₃₄D₄

C₂×C₂³⋊C₄

C₂×C₄²⋊C₂

C₂×C₂².D₄

C₂×C₂²⋊C₄

C₂×C₄⋊C₄

C₂²⋊C₄

C₂²×C₄

C₂³

C₂

# reps

Matrix representation of C₂⁴.₁₇₄C₂³ ►in GL₆(𝔽₅)

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

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

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

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

0	4	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	2	0
0	0	0	0	0	2

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

C₂⁴.₁₇₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{174}C_2^3

% in TeX

G:=Group("C2^4.174C2^3");

// GroupNames label

G:=SmallGroup(128,631);

// by ID

G=gap.SmallGroup(128,631);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,1018,2028]);

// Polycyclic

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

// generators/relations

G = C24.174C23 order 128 = 27

14th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₁₇₄C₂³ order 128 = 2⁷

14^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²