S3xC2^2xC4 - GroupNames

G = S₃xC₂²xC₄ order 96 = 2⁵·3

Direct product of C₂²xC₄ and S₃

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S₃xC₂²xC₄, C₁₂:₃C₂³, C₆.₂C₂⁴, D₆.₉C₂³, C₂³.₃₉D₆, Dic₃:₃C₂³, C₃:₁(C₂³xC₄), C₆:₁(C₂²xC₄), C₂.₁(S₃xC₂³), C₄ o(C₂²xDic₃), Dic₃ o(C₂²xC₄), (C₂²xC₁₂):₁₀C₂, (C₂xC₁₂):₁₄C₂², (S₃xC₂³).₃C₂, (C₂xC₆).₆₃C₂³, (C₂xDic₃):₁₂C₂², (C₂²xDic₃):₁₀C₂, C₂².₂₉(C₂²xS₃), (C₂²xC₆).₄₄C₂², (C₂²xS₃).₃₅C₂², C₄ o(S₃xC₂xC₄), (C₂xC₄)o(C₄xS₃), (C₂xC₆):₆(C₂xC₄), (C₂xC₄)o₂(C₂xDic₃), (C₂xC₄)o(C₂²xDic₃), (C₂²xC₄)o(C₂xDic₃), (C₂²xC₄)o(C₂²xDic₃), (C₂xC₄)o(S₃xC₂xC₄), SmallGroup(96,206)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃xC₂²xC₄

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂²xC₄

Lower central

C₃ — S₃xC₂²xC₄

Upper central

C₁ — C₂²xC₄

Generators and relations for S₃xC₂²xC₄
G = < a,b,c,d,e | a²=b²=c⁴=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 418 in 236 conjugacy classes, 145 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂⁴, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₂³xC₄, S₃xC₂xC₄, C₂²xDic₃, C₂²xC₁₂, S₃xC₂³, S₃xC₂²xC₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₆, C₂²xC₄, C₂⁴, C₄xS₃, C₂²xS₃, C₂³xC₄, S₃xC₂xC₄, S₃xC₂³, S₃xC₂²xC₄

Smallest permutation representation of S₃xC₂²xC₄
►On 48 points

Generators in S₄₈

(1 36)(2 33)(3 34)(4 35)(5 27)(6 28)(7 25)(8 26)(9 31)(10 32)(11 29)(12 30)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)

(1 24)(2 21)(3 22)(4 23)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 47)(34 48)(35 45)(36 46)

(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)

(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 47 10)(6 48 11)(7 45 12)(8 46 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 44 39)(34 41 40)(35 42 37)(36 43 38)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)(25 32)(26 29)(27 30)(28 31)(33 35)(34 36)(37 44)(38 41)(39 42)(40 43)(45 47)(46 48)

G:=sub<Sym(48)| (1,36)(2,33)(3,34)(4,35)(5,27)(6,28)(7,25)(8,26)(9,31)(10,32)(11,29)(12,30)(13,39)(14,40)(15,37)(16,38)(17,43)(18,44)(19,41)(20,42)(21,47)(22,48)(23,45)(24,46), (1,24)(2,21)(3,22)(4,23)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,47)(34,48)(35,45)(36,46), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,47,10)(6,48,11)(7,45,12)(8,46,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,44,39)(34,41,40)(35,42,37)(36,43,38), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,32)(26,29)(27,30)(28,31)(33,35)(34,36)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)>;

G:=Group( (1,36)(2,33)(3,34)(4,35)(5,27)(6,28)(7,25)(8,26)(9,31)(10,32)(11,29)(12,30)(13,39)(14,40)(15,37)(16,38)(17,43)(18,44)(19,41)(20,42)(21,47)(22,48)(23,45)(24,46), (1,24)(2,21)(3,22)(4,23)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,47)(34,48)(35,45)(36,46), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,47,10)(6,48,11)(7,45,12)(8,46,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,44,39)(34,41,40)(35,42,37)(36,43,38), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,32)(26,29)(27,30)(28,31)(33,35)(34,36)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48) );

G=PermutationGroup([[(1,36),(2,33),(3,34),(4,35),(5,27),(6,28),(7,25),(8,26),(9,31),(10,32),(11,29),(12,30),(13,39),(14,40),(15,37),(16,38),(17,43),(18,44),(19,41),(20,42),(21,47),(22,48),(23,45),(24,46)], [(1,24),(2,21),(3,22),(4,23),(5,39),(6,40),(7,37),(8,38),(9,43),(10,44),(11,41),(12,42),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(33,47),(34,48),(35,45),(36,46)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,47,10),(6,48,11),(7,45,12),(8,46,9),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,44,39),(34,41,40),(35,42,37),(36,43,38)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24),(25,32),(26,29),(27,30),(28,31),(33,35),(34,36),(37,44),(38,41),(39,42),(40,43),(45,47),(46,48)]])

S₃xC₂²xC₄ is a maximal subgroup of
C₂².₅₈(S₃xD₄) (C₂xC₄):₉D₁₂ D₆:C₄² D₆:(C₄:C₄) D₆:C₄:C₄ D₆:M₄(2) C₂⁴.₂₃D₆ C₄:(D₆:C₄) D₆:C₄:₆C₄ D₆:₆M₄(2) C₄²:₁₀D₆ C₄²:₁₄D₆ C₄:C₄:₂₁D₆ C₄:C₄:₂₆D₆ C₄:C₄:₂₈D₆ (C₂xD₄):₄₃D₆
S₃xC₂²xC₄ is a maximal quotient of
C₂⁴.₃₅D₆ C₆.₈2₊¹⁺⁴ C₄².₈₇D₆ C₄²:₉D₆ C₄².₁₈₈D₆ C₄².₉₁D₆ C₄²:₁₃D₆ C₄².₁₀₈D₆ C₄².₁₂₅D₆ C₄².₁₂₆D₆ M₄(2):₂₆D₆ M₄(2):₂₈D₆

48 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3	4A	···	4H	4I	···	4P	6A	···	6G	12A	···	12H
order	1	2	···	2	2	···	2	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	···	1	3	···	3	2	1	···	1	3	···	3	2	···	2	2	···	2

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	S₃	D₆	D₆	C₄xS₃
kernel	S₃xC₂²xC₄	S₃xC₂xC₄	C₂²xDic₃	C₂²xC₁₂	S₃xC₂³	C₂²xS₃	C₂²xC₄	C₂xC₄	C₂³	C₂²
# reps	1	12	1	1	1	16	1	6	1	8

Matrix representation of S₃xC₂²xC₄ ►in GL₄(F₁₃) generated by

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

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

8	0	0	0
0	5	0	0
0	0	8	0
0	0	0	8

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

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

G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,1,1] >;

S₃xC₂²xC₄ in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_4

% in TeX

G:=Group("S3xC2^2xC4");

// GroupNames label

G:=SmallGroup(96,206);

// by ID

G=gap.SmallGroup(96,206);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,69,2309]);

// Polycyclic

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

// generators/relations

G = S3xC22xC4 order 96 = 25·3

Direct product of C22xC4 and S3

G = S₃xC₂²xC₄ order 96 = 2⁵·3

Direct product of C₂²xC₄ and S₃