C2xS3xDic3 - GroupNames

G = C₂×S₃×Dic₃ order 144 = 2⁴·3²

Direct product of C₂, S₃ and Dic₃

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

Aliases: C₂×S₃×Dic₃, D₆.₉D₆, C₆².₆C₂², C₆⋊₃(C₄×S₃), (S₃×C₆)⋊₃C₄, C₂².₈S₃², C₆⋊₁(C₂×Dic₃), (C₂×C₆).₁₃D₆, (C₆×Dic₃)⋊₇C₂, C₃²⋊₃(C₂²×C₄), (C₂²×S₃).₂S₃, (S₃×C₆).₉C₂², C₆.₁₀(C₂²×S₃), (C₃×C₆).₁₀C₂³, C₃⋊Dic₃⋊₄C₂², C₃⋊₁(C₂²×Dic₃), (C₃×Dic₃)⋊₆C₂², C₃⋊₄(S₃×C₂×C₄), C₂.₂(C₂×S₃²), (C₃×C₆)⋊₂(C₂×C₄), (S₃×C₂×C₆).₃C₂, (C₃×S₃)⋊₂(C₂×C₄), (C₂×C₃⋊Dic₃)⋊₄C₂, SmallGroup(144,146)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×S₃×Dic₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — S₃×C₆ — S₃×Dic₃ — C₂×S₃×Dic₃

Lower central

C₃² — C₂×S₃×Dic₃

Upper central

C₁ — C₂²

Generators and relations for C₂×S₃×Dic₃
G = < a,b,c,d,e | a²=b³=c²=d⁶=1, e²=d³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 288 in 116 conjugacy classes, 56 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₃×Dic₃, C₃⋊Dic₃, S₃×C₆, C₆², S₃×C₂×C₄, C₂²×Dic₃, S₃×Dic₃, C₆×Dic₃, C₂×C₃⋊Dic₃, S₃×C₂×C₆, C₂×S₃×Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, C₂²×C₄, C₄×S₃, C₂×Dic₃, C₂²×S₃, S₃², S₃×C₂×C₄, C₂²×Dic₃, S₃×Dic₃, C₂×S₃², C₂×S₃×Dic₃

Smallest permutation representation of C₂×S₃×Dic₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

C₂×S₃×Dic₃ is a maximal subgroup of
C₆².₄₇C₂³ C₆².₄₈C₂³ C₆².₄₉C₂³ Dic₃⋊₄D₁₂ C₆².₅₁C₂³ C₆².₅₄C₂³ C₆².₅₅C₂³ Dic₃⋊D₁₂ D₆⋊₁Dic₆ D₆.D₁₂ D₆.₉D₁₂ D₆⋊₂Dic₆ D₆⋊₃Dic₆ D₁₂⋊Dic₃ D₆⋊₄Dic₆ C₆².₇₂C₂³ C₆².₁₁₁C₂³ C₆².₁₁₂C₂³ C₆².₁₁₃C₂³ C₆².₁₁₅C₂³ S₃²×C₂×C₄
C₂×S₃×Dic₃ is a maximal quotient of
D₁₂.₂Dic₃ D₁₂.Dic₃ C₆².₁₁C₂³ C₆².₁₃C₂³ C₆².₂₅C₂³ D₁₂⋊Dic₃ C₆².₉₇C₂³ C₆².₁₁₅C₂³

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	6H	6I	6J	6K	6L	6M	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	3	3	3	4	4	4	4	4	4	4	4	6	···	6	6	6	6	6	6	6	6	12	12	12	12
size	1	1	1	1	3	3	3	3	2	2	4	3	3	3	3	9	9	9	9	2	···	2	4	4	4	6	6	6	6	6	6	6	6

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+		+	+	+	-	+	+		+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₄	S₃	S₃	D₆	Dic₃	D₆	D₆	C₄×S₃	S₃²	S₃×Dic₃	C₂×S₃²
kernel	C₂×S₃×Dic₃	S₃×Dic₃	C₆×Dic₃	C₂×C₃⋊Dic₃	S₃×C₂×C₆	S₃×C₆	C₂×Dic₃	C₂²×S₃	Dic₃	D₆	D₆	C₂×C₆	C₆	C₂²	C₂	C₂
# reps	1	4	1	1	1	8	1	1	2	4	2	2	4	1	2	1

Matrix representation of C₂×S₃×Dic₃ ►in GL₆(𝔽₁₃)

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

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

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

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

5	0	0	0	0	0
0	5	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂×S₃×Dic₃ in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_3

% in TeX

G:=Group("C2xS3xDic3");

// GroupNames label

G:=SmallGroup(144,146);

// by ID

G=gap.SmallGroup(144,146);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,490,3461]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2×S3×Dic3 order 144 = 24·32

Direct product of C2, S3 and Dic3

G = C₂×S₃×Dic₃ order 144 = 2⁴·3²

Direct product of C₂, S₃ and Dic₃