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, (C₃xD₄):₁₅D₄, (C₃xQ₈):₁₅D₄, D₄:D₆:₅C₂, C₁₂:₃D₄:₈C₂, C₃:₅(D₄:₄D₄), D₄:₇(C₃:D₄), C₄oD₄.₂₅D₆, Q₈:₈(C₃:D₄), (C₂xD₄).₈₂D₆, C₆.₈₀C₂²wrC₂, C₁₂.₂₁₇(C₂xD₄), (C₂²xC₆).₂₄D₄, C₁₂.D₄:₁₁C₂, (C₂xD₁₂):₁₅C₂², (C₂xC₁₂).₂₁C₂³, (C₄xDic₃):₈C₂², Q₈:₃Dic₃:₁₁C₂, (C₆xD₄).₁₀₇C₂², C₂³.₁₂(C₃:D₄), C₄.Dic₃:₁₀C₂², (C₃x2₊¹⁺⁴):₁C₂, C₂.₁₄(C₂⁴:₄S₃), (C₂xC₆).₄₂(C₂xD₄), C₄.₆₄(C₂xC₃:D₄), (C₂xC₄).₂₁(C₂²xS₃), C₂².₁₄(C₂xC₃:D₄), (C₃xC₄oD₄).₁₉C₂², SmallGroup(192,800)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₂ — 2₊¹⁺⁴:₆S₃

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₂xD₁₂ — D₄:D₆ — 2₊¹⁺⁴:₆S₃

Lower central

C₃ — C₆ — C₂xC₁₂ — 2₊¹⁺⁴:₆S₃

Upper central

C₁ — C₂ — C₂xC₄ — 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, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², M₄(2), D₈, SD₁₆, C₂xD₄, C₂xD₄, C₄oD₄, C₄oD₄, C₃:C₈, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄.D₄, C₄wrC₂, C₄:₁D₄, C₈:C₂², 2₊¹⁺⁴, C₄.Dic₃, C₄xDic₃, D₄:S₃, Q₈:₂S₃, C₂xD₁₂, C₂xC₃:D₄, C₆xD₄, C₆xD₄, C₃xC₄oD₄, C₃xC₄oD₄, D₄:₄D₄, C₁₂.D₄, Q₈:₃Dic₃, C₁₂:₃D₄, D₄:D₆, C₃x2₊¹⁺⁴, 2₊¹⁺⁴:₆S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₂²wrC₂, C₂xC₃: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

C₃x2₊¹⁺⁴

2₊¹⁺⁴

C₃xD₄

C₃xQ₈

C₂²xC₆

C₂xD₄

C₄oD₄

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