ES+(2,2):7S3 - GroupNames

G = 2₊¹⁺⁴⋊₇S₃ order 192 = 2⁶·3

2^nd semidirect product of 2₊¹⁺⁴ and S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₇S₃, C₃⋊₃C₂≀C₂², (C₂×C₁₂)⋊₃D₄, (C₂²×C₆)⋊₄D₄, (C₂×D₄).₈₅D₆, C₂³⋊₂D₆⋊₁₉C₂, C₆.₈₃C₂²≀C₂, C₂³⋊₂(C₃⋊D₄), (S₃×C₂³)⋊₂C₂², (C₂²×C₆).₆C₂³, (C₆×D₄).₁₇₉C₂², C₂³.₇D₆⋊₁₀C₂, C₆.D₄⋊₇C₂², C₂³.₁₆(C₂²×S₃), (C₃×2₊¹⁺⁴)⋊₆C₂, C₂.₁₇(C₂⁴⋊₄S₃), (C₂×C₄)⋊₂(C₃⋊D₄), (C₂×C₆).₄₅(C₂×D₄), C₂².₁₇(C₂×C₃⋊D₄), SmallGroup(192,803)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — 2₊¹⁺⁴⋊₇S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — S₃×C₂³ — C₂³⋊₂D₆ — 2₊¹⁺⁴⋊₇S₃

Lower central

C₃ — C₆ — C₂²×C₆ — 2₊¹⁺⁴⋊₇S₃

Upper central

C₁ — C₂ — C₂³ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴⋊₇S₃
G = < a,b,c,d,e,f | a⁴=b²=d²=e³=f²=1, c²=a², bab=a^-1, ac=ca, ad=da, ae=ea, faf=a^-1cd, fcf=bc=cb, fdf=bd=db, be=eb, bf=fb, dcd=a²c, ce=ec, de=ed, fef=e^-1 >

Subgroups: 680 in 198 conjugacy classes, 43 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, D₆⋊C₄, C₆.D₄, C₂×C₃⋊D₄, C₆×D₄, C₆×D₄, C₃×C₄○D₄, S₃×C₂³, C₂≀C₂², C₂³.₇D₆, C₂³⋊₂D₆, C₃×2₊¹⁺⁴, 2₊¹⁺⁴⋊₇S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×C₃⋊D₄, C₂≀C₂², C₂⁴⋊₄S₃, 2₊¹⁺⁴⋊₇S₃

Permutation representations of 2₊¹⁺⁴⋊₇S₃
►On 24 points - transitive group 24T334

Generators in S₂₄

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(2 4)(5 7)(10 12)(13 15)(18 20)(21 23)

(1 22 3 24)(2 23 4 21)(5 20 7 18)(6 17 8 19)(9 14 11 16)(10 15 12 13)

(1 24)(2 21)(3 22)(4 23)(5 20)(6 17)(7 18)(8 19)(9 14)(10 15)(11 16)(12 13)

(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 21 10)(6 22 11)(7 23 12)(8 24 9)

(5 12)(6 11)(7 10)(8 9)(13 18)(14 19)(15 20)(16 17)(21 23)

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

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

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

G:=TransitiveGroup(24,334);

►On 24 points - transitive group 24T349

Generators in S₂₄

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)

(1 23 3 21)(2 24 4 22)(5 19 7 17)(6 20 8 18)(9 13 11 15)(10 14 12 16)

(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)

(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)

(1 3)(2 4)(5 11)(6 10)(7 9)(8 12)(13 20)(14 17)(15 18)(16 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,349);

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3	4A	4B	4C	4D	4E	4F	6A	6B	···	6J	12A	···	12F
order	1	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	6	6	···	6	12	···	12
size	1	1	2	2	2	4	4	4	12	12	2	4	4	4	24	24	24	2	4	···	4	4	···	4

33 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃⋊D₄

C₂≀C₂²

2₊¹⁺⁴⋊₇S₃

kernel

2₊¹⁺⁴⋊₇S₃

C₂³.₇D₆

C₂³⋊₂D₆

C₃×2₊¹⁺⁴

2₊¹⁺⁴

C₂×C₁₂

C₂²×C₆

C₂×D₄

C₂×C₄

C₂³

C₃

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊₇S₃ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	12	0	0	0	0
0	0	0	1	0	0
0	0	12	0	0	0
0	0	0	0	0	12
0	0	0	0	1	0

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	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	0	12	0	0
0	0	12	0	0	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	0	1	0	0
0	0	1	0	0	0

3	0	0	0	0	0
0	9	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

0	4	0	0	0	0
10	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	12
0	0	0	0	12	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,10,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

2₊¹⁺⁴⋊₇S₃ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_7S_3

% in TeX

G:=Group("ES+(2,2):7S3");

// GroupNames label

G:=SmallGroup(192,803);

// by ID

G=gap.SmallGroup(192,803);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,570,438,6278]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^3=f^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1*c*d,f*c*f=b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊7S3 order 192 = 26·3

2nd semidirect product of 2+ 1+4 and S3 acting via S3/C3=C2

G = 2₊¹⁺⁴⋊₇S₃ order 192 = 2⁶·3

2^nd semidirect product of 2₊¹⁺⁴ and S₃ acting via S₃/C₃=C₂