S3xES+(2,2)

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S₃×2₊⁽¹⁺⁴⁾, C₆.₁₅C₂⁵, D₁₂⋊₁₃C₂³, C₁₂.₅₀C₂⁴, D₆.₁₄C₂⁴, Dic₆⋊₁₂C₂³, Dic₃.₁₀C₂⁴, C₄○D₄⋊₁₃D₆, (C₂×D₄)⋊₃₁D₆, D₄○D₁₂⋊₁₁C₂, D₄⋊₆D₆⋊₉C₂, (C₄×S₃)⋊₃C₂³, (C₂×C₁₂)⋊₂C₂³, C₃⋊D₄⋊₆C₂³, (C₂×C₆).₆C₂⁴, (S₃×D₄)⋊₁₇C₂², (C₆×D₄)⋊₂₅C₂², D₄⋊₁₀(C₂²×S₃), (C₃×D₄)⋊₁₁C₂³, C₄.₄₇(S₃×C₂³), C₂.₁₆(S₃×C₂⁴), C₂³⋊₃(C₂²×S₃), (C₂²×C₆)⋊₂C₂³, (C₃×Q₈)⋊₁₀C₂³, Q₈⋊₁₀(C₂²×S₃), (S₃×Q₈)⋊₂₀C₂², C₃⋊₃(C₂×2₊⁽¹⁺⁴⁾), C₄○D₁₂⋊₁₂C₂², (C₂×D₁₂)⋊₄₀C₂², (C₂²×S₃)⋊₆C₂³, (C₂×Dic₃)⋊₆C₂³, C₂².₃(S₃×C₂³), D₄⋊₂S₃⋊₁₅C₂², (S₃×C₂³)⋊₁₉C₂², Q₈⋊₃S₃⋊₁₅C₂², (C₃×2₊⁽¹⁺⁴⁾)⋊₄C₂, (C₂×S₃×D₄)⋊₂₉C₂, (S₃×C₄○D₄)⋊₇C₂, (S₃×C₂×C₄)⋊₃₆C₂², (C₂×C₄)⋊₂(C₂²×S₃), (C₃×C₄○D₄)⋊₁₀C₂², (C₂×C₃⋊D₄)⋊₃₂C₂², SmallGroup(192,1524)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃×2₊⁽¹⁺⁴⁾

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂³ — C₂×S₃×D₄ — S₃×2₊⁽¹⁺⁴⁾

Lower central

C₃ — C₆ — S₃×2₊⁽¹⁺⁴⁾

Upper central

C₁ — C₂ — 2₊⁽¹⁺⁴⁾

Subgroups: 2152 in 898 conjugacy classes, 445 normal (8 characteristic)
C₁, C₂, C₂^[×20], C₃, C₄^[×6], C₄^[×6], C₂²^[×9], C₂²^[×52], S₃^[×2], S₃^[×9], C₆, C₆^[×9], C₂×C₄^[×9], C₂×C₄^[×33], D₄^[×18], D₄^[×54], Q₈^[×2], Q₈^[×6], C₂³^[×6], C₂³^[×39], Dic₃^[×6], C₁₂^[×6], D₆, D₆^[×9], D₆^[×36], C₂×C₆^[×9], C₂×C₆^[×6], C₂²×C₄^[×9], C₂×D₄^[×9], C₂×D₄^[×81], C₂×Q₈^[×2], C₄○D₄^[×6], C₄○D₄^[×42], C₂⁴^[×6], Dic₆^[×6], C₄×S₃^[×24], D₁₂^[×18], C₂×Dic₃^[×9], C₃⋊D₄^[×36], C₂×C₁₂^[×9], C₃×D₄^[×18], C₃×Q₈^[×2], C₂²×S₃^[×27], C₂²×S₃^[×12], C₂²×C₆^[×6], C₂²×D₄^[×9], C₂×C₄○D₄^[×6], 2₊⁽¹⁺⁴⁾, 2₊⁽¹⁺⁴⁾^[×15], S₃×C₂×C₄^[×9], C₂×D₁₂^[×9], C₄○D₁₂^[×18], S₃×D₄^[×54], D₄⋊₂S₃^[×18], S₃×Q₈^[×2], Q₈⋊₃S₃^[×6], C₂×C₃⋊D₄^[×18], C₆×D₄^[×9], C₃×C₄○D₄^[×6], S₃×C₂³^[×6], C₂×2₊⁽¹⁺⁴⁾, C₂×S₃×D₄^[×9], D₄⋊₆D₆^[×9], S₃×C₄○D₄^[×6], D₄○D₁₂^[×6], C₃×2₊⁽¹⁺⁴⁾, S₃×2₊⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×31], C₂²^[×155], S₃, C₂³^[×155], D₆^[×15], C₂⁴^[×31], C₂²×S₃^[×35], 2₊⁽¹⁺⁴⁾^[×2], C₂⁵, S₃×C₂³^[×15], C₂×2₊⁽¹⁺⁴⁾, S₃×C₂⁴, S₃×2₊⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e,f | a³=b²=c⁴=d²=f²=1, e²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=c²e >

Permutation representations
►On 24 points - transitive group 24T335

Generators in S₂₄

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

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

(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)(6 8)(9 11)(13 15)(18 20)(22 24)

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

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

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

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

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

G:=TransitiveGroup(24,335);

Matrix representation ►G ⊆ GL₆(ℤ)

-1	-1	0	0	0	0
1	0	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
1	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	1

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	2	0	0
0	0	-1	1	0	0
0	0	0	-1	0	-1
0	0	1	-1	1	0

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	1	0	0	0
0	0	1	-1	0	0
0	0	0	0	1	0
0	0	-1	0	0	-1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	2	0
0	0	0	0	1	-1
0	0	-1	0	-1	0
0	0	-1	1	-1	0

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	-1	0	-1	0
0	0	-1	0	0	-1

G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,0,1,0,0,2,1,-1,-1,0,0,0,0,0,1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,-1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,0,0,1,0,0,2,1,-1,-1,0,0,0,-1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

51 conjugacy classes

class	1	2A	2B	···	2J	2K	2L	2M	···	2U	3	4A	···	4F	4G	···	4L	6A	6B	···	6J	12A	···	12F
order	1	2	2	···	2	2	2	2	···	2	3	4	···	4	4	···	4	6	6	···	6	12	···	12
size	1	1	2	···	2	3	3	6	···	6	2	2	···	2	6	···	6	2	4	···	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	2₊⁽¹⁺⁴⁾	S₃×2₊⁽¹⁺⁴⁾
kernel	S₃×2₊⁽¹⁺⁴⁾	C₂×S₃×D₄	D₄⋊₆D₆	S₃×C₄○D₄	D₄○D₁₂	C₃×2₊⁽¹⁺⁴⁾	2₊⁽¹⁺⁴⁾	C₂×D₄	C₄○D₄	S₃	C₁
# reps	1	9	9	6	6	1	1	9	6	2	1

In GAP, Magma, Sage, TeX

S_3\times 2_+^{(1+4)}

% in TeX

G:=Group("S3xES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1524);

// by ID

G=gap.SmallGroup(192,1524);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,851,6278]);

// Polycyclic

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

// generators/relations

G = S₃×2₊⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of S₃ and 2₊⁽¹⁺⁴⁾

G = S3×2+ (1+4) order 192 = 26·3

Direct product of S3 and 2+ (1+4)

G = S₃×2₊⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of S₃ and 2₊⁽¹⁺⁴⁾