ES+(2,2):6S3 - GroupNames

G = 2₊¹⁺⁴⋊₆S₃ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₆S₃, (C₃×D₄)⋊₁₅D₄, (C₃×Q₈)⋊₁₅D₄, D₄⋊D₆⋊₅C₂, C₁₂⋊₃D₄⋊₈C₂, C₃⋊₅(D₄⋊₄D₄), D₄⋊₇(C₃⋊D₄), C₄○D₄.₂₅D₆, Q₈⋊₈(C₃⋊D₄), (C₂×D₄).₈₂D₆, C₆.₈₀C₂²≀C₂, C₁₂.₂₁₇(C₂×D₄), (C₂²×C₆).₂₄D₄, C₁₂.D₄⋊₁₁C₂, (C₂×D₁₂)⋊₁₅C₂², (C₂×C₁₂).₂₁C₂³, (C₄×Dic₃)⋊₈C₂², Q₈⋊₃Dic₃⋊₁₁C₂, (C₆×D₄).₁₀₇C₂², C₂³.₁₂(C₃⋊D₄), C₄.Dic₃⋊₁₀C₂², (C₃×2₊¹⁺⁴)⋊₁C₂, C₂.₁₄(C₂⁴⋊₄S₃), (C₂×C₆).₄₂(C₂×D₄), C₄.₆₄(C₂×C₃⋊D₄), (C₂×C₄).₂₁(C₂²×S₃), C₂².₁₄(C₂×C₃⋊D₄), (C₃×C₄○D₄).₁₉C₂², SmallGroup(192,800)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — 2₊¹⁺⁴⋊₆S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₂×D₁₂ — D₄⋊D₆ — 2₊¹⁺⁴⋊₆S₃

Lower central

C₃ — C₆ — C₂×C₁₂ — 2₊¹⁺⁴⋊₆S₃

Upper central

C₁ — C₂ — C₂×C₄ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴⋊₆S₃
G = < a,b,c,d,e,f | a⁴=b²=d²=e³=f²=1, c²=a², bab=faf=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a²c, ce=ec, de=ed, fdf=cd, fef=e^-1 >

Subgroups: 520 in 168 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₆, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, C₃⋊C₈, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₄.Dic₃, C₄×Dic₃, D₄⋊S₃, Q₈⋊₂S₃, C₂×D₁₂, C₂×C₃⋊D₄, C₆×D₄, C₆×D₄, C₃×C₄○D₄, C₃×C₄○D₄, D₄⋊₄D₄, C₁₂.D₄, Q₈⋊₃Dic₃, C₁₂⋊₃D₄, D₄⋊D₆, C₃×2₊¹⁺⁴, 2₊¹⁺⁴⋊₆S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂²≀C₂, C₂×C₃⋊D₄, D₄⋊₄D₄, C₂⁴⋊₄S₃, 2₊¹⁺⁴⋊₆S₃

Permutation representations of 2₊¹⁺⁴⋊₆S₃
►On 24 points - transitive group 24T347

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

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

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

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

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

G:=TransitiveGroup(24,347);

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	6A	6B	···	6J	8A	8B	12A	···	12F
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	6	6	···	6	8	8	12	···	12
size	1	1	2	4	4	4	4	24	2	2	2	4	4	12	12	2	4	···	4	24	24	4	···	4

33 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃⋊D₄

D₄⋊₄D₄

2₊¹⁺⁴⋊₆S₃

kernel

2₊¹⁺⁴⋊₆S₃

C₁₂.D₄

Q₈⋊₃Dic₃

C₁₂⋊₃D₄

D₄⋊D₆

C₃×2₊¹⁺⁴

2₊¹⁺⁴

C₃×D₄

C₃×Q₈

C₂²×C₆

C₂×D₄

C₄○D₄

D₄

Q₈

C₂³

C₃

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊₆S₃ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	2	0	0
0	0	72	1	0	0
0	0	0	0	72	2
0	0	0	0	72	1

1	72	0	0	0	0
0	72	0	0	0	0
0	0	0	0	72	2
0	0	0	0	0	1
0	0	72	2	0	0
0	0	0	1	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	71	0	0
0	0	1	72	0	0
0	0	0	0	72	2
0	0	0	0	72	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	72	2
0	0	0	0	72	1
0	0	1	71	0	0
0	0	1	72	0	0

64	45	0	0	0	0
0	8	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

1	0	0	0	0	0
2	72	0	0	0	0
0	0	1	0	0	0
0	0	1	72	0	0
0	0	0	0	1	71
0	0	0	0	0	72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,2,1,0,0,72,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,72,72,0,0,0,0,2,1,0,0],[64,0,0,0,0,0,45,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72] >;

2₊¹⁺⁴⋊₆S₃ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_6S_3

% in TeX

G:=Group("ES+(2,2):6S3");

// GroupNames label

G:=SmallGroup(192,800);

// by ID

G=gap.SmallGroup(192,800);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,570,1684,851,438,102,6278]);

// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=c*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊6S3 order 192 = 26·3

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

G = 2₊¹⁺⁴⋊₆S₃ order 192 = 2⁶·3

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