S3xC3:Dic3 - GroupNames

G = S₃×C₃⋊Dic₃ order 216 = 2³·3³

Direct product of S₃ and C₃⋊Dic₃

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

Aliases: S₃×C₃⋊Dic₃, C₆.₉S₃², D₆.(C₃⋊S₃), (C₃×S₃)⋊Dic₃, C₃³⋊₆(C₂×C₄), (S₃×C₆).₅S₃, C₃⋊₃(S₃×Dic₃), (C₃×C₆).₄₃D₆, (S₃×C₃²)⋊₃C₄, C₃²⋊₁₃(C₄×S₃), C₃³⋊₅C₄⋊₁C₂, C₃²⋊₅(C₂×Dic₃), (C₃²×C₆).₆C₂², C₆.₁(C₂×C₃⋊S₃), C₂.₁(S₃×C₃⋊S₃), (S₃×C₃×C₆).₂C₂, C₃⋊₁(C₂×C₃⋊Dic₃), (C₃×C₃⋊Dic₃)⋊₄C₂, SmallGroup(216,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×C₃⋊Dic₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — S₃×C₃×C₆ — S₃×C₃⋊Dic₃

Lower central

C₃³ — S₃×C₃⋊Dic₃

Upper central

C₁ — C₂

Generators and relations for S₃×C₃⋊Dic₃
G = < a,b,c,d,e | a³=b²=c³=d⁶=1, e²=d³, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 444 in 120 conjugacy classes, 46 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₃², C₃², C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, S₃×C₆, C₆², S₃×C₃², C₃²×C₆, S₃×Dic₃, C₂×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, S₃×C₃×C₆, S₃×C₃⋊Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, C₃⋊S₃, C₄×S₃, C₂×Dic₃, C₃⋊Dic₃, S₃², C₂×C₃⋊S₃, S₃×Dic₃, C₂×C₃⋊Dic₃, S₃×C₃⋊S₃, S₃×C₃⋊Dic₃

Smallest permutation representation of S₃×C₃⋊Dic₃
►On 72 points

Generators in S₇₂

(1 24 25)(2 19 26)(3 20 27)(4 21 28)(5 22 29)(6 23 30)(7 71 14)(8 72 15)(9 67 16)(10 68 17)(11 69 18)(12 70 13)(31 45 39)(32 46 40)(33 47 41)(34 48 42)(35 43 37)(36 44 38)(49 59 66)(50 60 61)(51 55 62)(52 56 63)(53 57 64)(54 58 65)

(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 42)(14 37)(15 38)(16 39)(17 40)(18 41)(19 51)(20 52)(21 53)(22 54)(23 49)(24 50)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)

(1 20 29)(2 21 30)(3 22 25)(4 23 26)(5 24 27)(6 19 28)(7 67 18)(8 68 13)(9 69 14)(10 70 15)(11 71 16)(12 72 17)(31 41 43)(32 42 44)(33 37 45)(34 38 46)(35 39 47)(36 40 48)(49 62 57)(50 63 58)(51 64 59)(52 65 60)(53 66 55)(54 61 56)

(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)

(1 39 4 42)(2 38 5 41)(3 37 6 40)(7 66 10 63)(8 65 11 62)(9 64 12 61)(13 60 16 57)(14 59 17 56)(15 58 18 55)(19 36 22 33)(20 35 23 32)(21 34 24 31)(25 45 28 48)(26 44 29 47)(27 43 30 46)(49 68 52 71)(50 67 53 70)(51 72 54 69)

G:=sub<Sym(72)| (1,24,25)(2,19,26)(3,20,27)(4,21,28)(5,22,29)(6,23,30)(7,71,14)(8,72,15)(9,67,16)(10,68,17)(11,69,18)(12,70,13)(31,45,39)(32,46,40)(33,47,41)(34,48,42)(35,43,37)(36,44,38)(49,59,66)(50,60,61)(51,55,62)(52,56,63)(53,57,64)(54,58,65), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,42)(14,37)(15,38)(16,39)(17,40)(18,41)(19,51)(20,52)(21,53)(22,54)(23,49)(24,50)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (1,20,29)(2,21,30)(3,22,25)(4,23,26)(5,24,27)(6,19,28)(7,67,18)(8,68,13)(9,69,14)(10,70,15)(11,71,16)(12,72,17)(31,41,43)(32,42,44)(33,37,45)(34,38,46)(35,39,47)(36,40,48)(49,62,57)(50,63,58)(51,64,59)(52,65,60)(53,66,55)(54,61,56), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,39,4,42)(2,38,5,41)(3,37,6,40)(7,66,10,63)(8,65,11,62)(9,64,12,61)(13,60,16,57)(14,59,17,56)(15,58,18,55)(19,36,22,33)(20,35,23,32)(21,34,24,31)(25,45,28,48)(26,44,29,47)(27,43,30,46)(49,68,52,71)(50,67,53,70)(51,72,54,69)>;

G:=Group( (1,24,25)(2,19,26)(3,20,27)(4,21,28)(5,22,29)(6,23,30)(7,71,14)(8,72,15)(9,67,16)(10,68,17)(11,69,18)(12,70,13)(31,45,39)(32,46,40)(33,47,41)(34,48,42)(35,43,37)(36,44,38)(49,59,66)(50,60,61)(51,55,62)(52,56,63)(53,57,64)(54,58,65), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,42)(14,37)(15,38)(16,39)(17,40)(18,41)(19,51)(20,52)(21,53)(22,54)(23,49)(24,50)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (1,20,29)(2,21,30)(3,22,25)(4,23,26)(5,24,27)(6,19,28)(7,67,18)(8,68,13)(9,69,14)(10,70,15)(11,71,16)(12,72,17)(31,41,43)(32,42,44)(33,37,45)(34,38,46)(35,39,47)(36,40,48)(49,62,57)(50,63,58)(51,64,59)(52,65,60)(53,66,55)(54,61,56), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,39,4,42)(2,38,5,41)(3,37,6,40)(7,66,10,63)(8,65,11,62)(9,64,12,61)(13,60,16,57)(14,59,17,56)(15,58,18,55)(19,36,22,33)(20,35,23,32)(21,34,24,31)(25,45,28,48)(26,44,29,47)(27,43,30,46)(49,68,52,71)(50,67,53,70)(51,72,54,69) );

G=PermutationGroup([[(1,24,25),(2,19,26),(3,20,27),(4,21,28),(5,22,29),(6,23,30),(7,71,14),(8,72,15),(9,67,16),(10,68,17),(11,69,18),(12,70,13),(31,45,39),(32,46,40),(33,47,41),(34,48,42),(35,43,37),(36,44,38),(49,59,66),(50,60,61),(51,55,62),(52,56,63),(53,57,64),(54,58,65)], [(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,42),(14,37),(15,38),(16,39),(17,40),(18,41),(19,51),(20,52),(21,53),(22,54),(23,49),(24,50),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)], [(1,20,29),(2,21,30),(3,22,25),(4,23,26),(5,24,27),(6,19,28),(7,67,18),(8,68,13),(9,69,14),(10,70,15),(11,71,16),(12,72,17),(31,41,43),(32,42,44),(33,37,45),(34,38,46),(35,39,47),(36,40,48),(49,62,57),(50,63,58),(51,64,59),(52,65,60),(53,66,55),(54,61,56)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,39,4,42),(2,38,5,41),(3,37,6,40),(7,66,10,63),(8,65,11,62),(9,64,12,61),(13,60,16,57),(14,59,17,56),(15,58,18,55),(19,36,22,33),(20,35,23,32),(21,34,24,31),(25,45,28,48),(26,44,29,47),(27,43,30,46),(49,68,52,71),(50,67,53,70),(51,72,54,69)]])

S₃×C₃⋊Dic₃ is a maximal subgroup of
C₃³⋊M₄(2) S₃²×Dic₃ D₆.₄S₃² D₆⋊S₃⋊S₃ D₆.₆S₃² (C₃×D₁₂)⋊S₃ D₁₂⋊(C₃⋊S₃) C₄×S₃×C₃⋊S₃ C₆².₉₀D₆ C₆².₉₁D₆
S₃×C₃⋊Dic₃ is a maximal quotient of
C₃³⋊₇M₄(2) C₆².₇₇D₆ C₆².₈₀D₆

36 conjugacy classes

class	1	2A	2B	2C	3A	···	3E	3F	3G	3H	3I	4A	4B	4C	4D	6A	···	6E	6F	6G	6H	6I	6J	···	6Q	12A	12B
order	1	2	2	2	3	···	3	3	3	3	3	4	4	4	4	6	···	6	6	6	6	6	6	···	6	12	12
size	1	1	3	3	2	···	2	4	4	4	4	9	9	27	27	2	···	2	4	4	4	4	6	···	6	18	18

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+		+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₄	S₃	S₃	Dic₃	D₆	C₄×S₃	S₃²	S₃×Dic₃
kernel	S₃×C₃⋊Dic₃	C₃×C₃⋊Dic₃	C₃³⋊₅C₄	S₃×C₃×C₆	S₃×C₃²	C₃⋊Dic₃	S₃×C₆	C₃×S₃	C₃×C₆	C₃²	C₆	C₃
# reps	1	1	1	1	4	1	4	8	5	2	4	4

Matrix representation of S₃×C₃⋊Dic₃ ►in GL₆(𝔽₁₃)

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	12	1
0	0	0	0	12	0

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	0	12
0	0	0	0	12	0

12	1	0	0	0	0
12	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

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

0	12	0	0	0	0
12	0	0	0	0	0
0	0	1	0	0	0
0	0	12	12	0	0
0	0	0	0	5	0
0	0	0	0	0	5

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

S₃×C₃⋊Dic₃ in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes {\rm Dic}_3

% in TeX

G:=Group("S3xC3:Dic3");

// GroupNames label

G:=SmallGroup(216,124);

// by ID

G=gap.SmallGroup(216,124);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,201,730,5189]);

// Polycyclic

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

// generators/relations

G = S3×C3⋊Dic3 order 216 = 23·33

Direct product of S3 and C3⋊Dic3

G = S₃×C₃⋊Dic₃ order 216 = 2³·3³

Direct product of S₃ and C₃⋊Dic₃