C2xS3xSL(2,3) - GroupNames

G = C₂xS₃xSL₂(F₃) order 288 = 2⁵·3²

Direct product of C₂, S₃ and SL₂(F₃)

Aliases: C₂xS₃xSL₂(F₃), Q₈:₂(S₃xC₆), (S₃xQ₈):₃C₆, (C₆xQ₈):₁C₆, D₆.₄(C₂xA₄), C₆:(C₂xSL₂(F₃)), C₂².₈(S₃xA₄), C₆.₅(C₂²xA₄), (C₂²xS₃).₄A₄, C₃:(C₂²xSL₂(F₃)), (C₆xSL₂(F₃)):₄C₂, (C₃xSL₂(F₃)):₆C₂², (C₂xS₃xQ₈):C₃, C₂.₆(C₂xS₃xA₄), (C₃xQ₈):(C₂xC₆), (C₂xQ₈):₄(C₃xS₃), (C₂xC₆).₁₉(C₂xA₄), SmallGroup(288,922)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₃xQ₈ — C₂xS₃xSL₂(F₃)

Chief series

C₁ — C₂ — C₆ — C₃xQ₈ — C₃xSL₂(F₃) — S₃xSL₂(F₃) — C₂xS₃xSL₂(F₃)

Lower central

C₃xQ₈ — C₂xS₃xSL₂(F₃)

Upper central

C₁ — C₂²

Generators and relations for C₂xS₃xSL₂(F₃)
G = < a,b,c,d,e,f | a²=b³=c²=d⁴=f³=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede^-1=d^-1, fdf^-1=e, fef^-1=de >

Subgroups: 550 in 129 conjugacy classes, 35 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, Q₈, Q₈, C₂³, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₃xS₃, C₃xC₆, SL₂(F₃), SL₂(F₃), Dic₆, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₃xQ₈, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xQ₈, S₃xC₆, C₆², C₂xSL₂(F₃), C₂xSL₂(F₃), C₂xDic₆, S₃xC₂xC₄, S₃xQ₈, S₃xQ₈, C₆xQ₈, C₃xSL₂(F₃), S₃xC₂xC₆, C₂²xSL₂(F₃), C₂xS₃xQ₈, S₃xSL₂(F₃), C₆xSL₂(F₃), C₂xS₃xSL₂(F₃)
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, A₄, D₆, C₂xC₆, C₃xS₃, SL₂(F₃), C₂xA₄, S₃xC₆, C₂xSL₂(F₃), C₂²xA₄, S₃xA₄, C₂²xSL₂(F₃), S₃xSL₂(F₃), C₂xS₃xA₄, C₂xS₃xSL₂(F₃)

Smallest permutation representation of C₂xS₃xSL₂(F₃)
►On 48 points

Generators in S₄₈

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

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

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

(2 23 24)(4 21 22)(5 6 39)(7 8 37)(10 17 18)(12 19 20)(13 25 26)(15 27 28)(29 41 42)(31 43 44)(34 45 46)(36 47 48)

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	6D	···	6I	6J	···	6O	6P	···	6W	12A	12B
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	6	6	6	6	···	6	6	···	6	6	···	6	12	12
size	1	1	1	1	3	3	3	3	2	4	4	8	8	6	6	18	18	2	2	2	4	···	4	8	···	8	12	···	12	12	12

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	4	4	6	6
type	+	+	+				+	+		-			+	+	+	-		+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃xS₃	SL₂(F₃)	SL₂(F₃)	S₃xC₆	A₄	C₂xA₄	C₂xA₄	S₃xSL₂(F₃)	S₃xSL₂(F₃)	S₃xA₄	C₂xS₃xA₄
kernel	C₂xS₃xSL₂(F₃)	S₃xSL₂(F₃)	C₆xSL₂(F₃)	C₂xS₃xQ₈	S₃xQ₈	C₆xQ₈	C₂xSL₂(F₃)	SL₂(F₃)	C₂xQ₈	D₆	D₆	Q₈	C₂²xS₃	D₆	C₂xC₆	C₂	C₂	C₂²	C₂
# reps	1	2	1	2	4	2	1	1	2	4	8	2	1	2	1	2	4	1	1

Matrix representation of C₂xS₃xSL₂(F₃) ►in GL₄(F₁₃) generated by

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

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

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

1	0	0	0
0	1	0	0
0	0	1	11
0	0	1	12

1	0	0	0
0	1	0	0
0	0	12	7
0	0	9	1

3	0	0	0
0	3	0	0
0	0	1	0
0	0	1	3

G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,11,12],[1,0,0,0,0,1,0,0,0,0,12,9,0,0,7,1],[3,0,0,0,0,3,0,0,0,0,1,1,0,0,0,3] >;

C₂xS₃xSL₂(F₃) in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm SL}_2({\mathbb F}_3)

% in TeX

G:=Group("C2xS3xSL(2,3)");

// GroupNames label

G:=SmallGroup(288,922);

// by ID

G=gap.SmallGroup(288,922);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,269,360,123,515,242,4037]);

// Polycyclic

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

// generators/relations

G = C2xS3xSL2(F3) order 288 = 25·32

Direct product of C2, S3 and SL2(F3)

G = C₂xS₃xSL₂(F₃) order 288 = 2⁵·3²

Direct product of C₂, S₃ and SL₂(F₃)