C2xC2^2.19C2^4 - GroupNames

G = C₂×C₂².₁₉C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₁₉C₂⁴

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

Aliases: C₂×C₂².₁₉C₂⁴, C₄²⋊₉C₂³, C₂³.₁₂C₂⁴, C₂².₂₄C₂⁵, C₂⁵.₉₆C₂², C₂⁴.₆₀₈C₂³, (C₂⁴×C₄)⋊₉C₂, C₄⋊C₄⋊₁₆C₂³, (C₂²×C₄)⋊₅₆D₄, C₂.₈(D₄×C₂³), (C₄×D₄)⋊₈₉C₂², (C₂×D₄)⋊₁₅C₂³, C₂³⋊₇(C₄○D₄), (C₂×C₄).₂₉C₂⁴, (C₂×Q₈)⋊₁₄C₂³, C₄⋊D₄⋊₉₅C₂², C₂²⋊C₄⋊₁₇C₂³, (C₂×C₄²)⋊₄₂C₂², (C₂²×C₄)⋊₁₂C₂³, (C₂³×C₄)⋊₂₈C₂², C₂³.₇₀₇(C₂×D₄), C₄.₁₆₉(C₂²×D₄), C₂²≀C₂⋊₄₁C₂², C₄ ○(C₂².₁₉C₂⁴), (C₂²×D₄)⋊₅₈C₂², C₂²⋊Q₈⋊₁₀₃C₂², (C₂²×Q₈)⋊₅₆C₂², C₂².₄₆(C₂²×D₄), C₄²⋊C₂⋊₈₃C₂², C₂².D₄⋊₆₆C₂², (C₂×C₄×D₄)⋊₆₉C₂, C₄ ○(C₂×C₄⋊D₄), C₄ ○(C₂×C₂²⋊Q₈), C₄ ○(C₂×C₂²≀C₂), (C₂×C₄)⋊₂₂(C₂×D₄), (C₂×C₄)○₂ C₂²≀C₂, C₂²⋊₁(C₂×C₄○D₄), (C₂×C₄)○₂(C₄⋊D₄), (C₂×C₄⋊D₄)⋊₇₆C₂, (C₂²×C₄)○C₂²≀C₂, (C₂×C₄)○₂(C₂²⋊Q₈), (C₂×C₂²⋊Q₈)⋊₈₆C₂, (C₂×C₂²≀C₂)⋊₃₀C₂, C₂.₈(C₂²×C₄○D₄), C₄ ○(C₂×C₂².D₄), (C₂×C₄⋊C₄)⋊₁₂₅C₂², (C₂²×C₄○D₄)⋊₁₁C₂, (C₂×C₄○D₄)⋊₆₅C₂², (C₂×C₄²⋊C₂)⋊₅₁C₂, (C₂²×C₄)○(C₂²⋊Q₈), (C₂×C₂²⋊C₄)⋊₈₃C₂², (C₂×C₄)○(C₂².₁₉C₂⁴), (C₂×C₄)○₂(C₂².D₄), (C₂×C₂².D₄)⋊₆₇C₂, (C₂×C₄)○₂(C₂×C₂²⋊Q₈), (C₂×C₄)○₂(C₂×C₂²≀C₂), (C₂²×C₄)○(C₂×C₂²⋊Q₈), (C₂²×C₄)○(C₂×C₂²≀C₂), SmallGroup(128,2167)

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

Derived series

C₁ — C₂² — C₂×C₂².₁₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂⁴×C₄ — C₂×C₂².₁₉C₂⁴

Lower central

C₁ — C₂² — C₂×C₂².₁₉C₂⁴

Upper central

C₁ — C₂²×C₄ — C₂×C₂².₁₉C₂⁴

Jennings

C₁ — C₂² — C₂×C₂².₁₉C₂⁴

Generators and relations for C₂×C₂².₁₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1452 in 940 conjugacy classes, 452 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂⁵, C₂×C₄²⋊C₂, C₂×C₄×D₄, C₂×C₂²≀C₂, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂⁴×C₄, C₂²×C₄○D₄, C₂×C₂².₁₉C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂⁵, C₂².₁₉C₂⁴, D₄×C₂³, C₂²×C₄○D₄, C₂×C₂².₁₉C₂⁴

Smallest permutation representation of C₂×C₂².₁₉C₂⁴
►On 32 points

Generators in S₃₂

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

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

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

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

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

(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)

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	2V	2W	4A	···	4H	4I	···	4T	4U	···	4AF
order	1	2	···	2	2	···	2	2	2	2	2	4	···	4	4	···	4	4	···	4
size	1	1	···	1	2	···	2	4	4	4	4	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

kernel

C₂×C₂².₁₉C₂⁴

C₂×C₄²⋊C₂

C₂×C₄×D₄

C₂×C₂²≀C₂

C₂×C₄⋊D₄

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂².₁₉C₂⁴

C₂⁴×C₄

C₂²×C₄○D₄

C₂²×C₄

C₂³

# reps

Matrix representation of C₂×C₂².₁₉C₂⁴ ►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

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	4	0
0	0	0	0	0	4

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

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,2,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂×C₂².₁₉C₂⁴ in GAP, Magma, Sage, TeX

C_2\times C_2^2._{19}C_2^4

% in TeX

G:=Group("C2xC2^2.19C2^4");

// GroupNames label

G:=SmallGroup(128,2167);

// by ID

G=gap.SmallGroup(128,2167);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,136]);

// Polycyclic

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

// generators/relations

G = C2×C22.19C24 order 128 = 27

Direct product of C2 and C22.19C24

G = C₂×C₂².₁₉C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₁₉C₂⁴