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## G = S3×C3×C12order 216 = 23·33

### Direct product of C3×C12 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C12
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C3×C12
 Lower central C3 — S3×C3×C12
 Upper central C1 — C3×C12

Generators and relations for S3×C3×C12
G = < a,b,c,d | a3=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 204 in 120 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C32, C32×C6, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C3×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, C12, D6, C2×C6, C3×S3, C3×C6, C4×S3, C2×C12, C3×C12, S3×C6, C62, S3×C32, S3×C12, C6×C12, S3×C3×C6, S3×C3×C12

Smallest permutation representation of S3×C3×C12
On 72 points
Generators in S72
(1 40 29)(2 41 30)(3 42 31)(4 43 32)(5 44 33)(6 45 34)(7 46 35)(8 47 36)(9 48 25)(10 37 26)(11 38 27)(12 39 28)(13 56 72)(14 57 61)(15 58 62)(16 59 63)(17 60 64)(18 49 65)(19 50 66)(20 51 67)(21 52 68)(22 53 69)(23 54 70)(24 55 71)
(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 48 33)(2 37 34)(3 38 35)(4 39 36)(5 40 25)(6 41 26)(7 42 27)(8 43 28)(9 44 29)(10 45 30)(11 46 31)(12 47 32)(13 64 52)(14 65 53)(15 66 54)(16 67 55)(17 68 56)(18 69 57)(19 70 58)(20 71 59)(21 72 60)(22 61 49)(23 62 50)(24 63 51)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 59)(26 60)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)

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

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

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

S3×C3×C12 is a maximal subgroup of   C337M4(2)  C12.73S32  C12.57S32  C12.58S32

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 12A ··· 12P 12Q ··· 12AH 12AI ··· 12AX order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 ··· 1 2 ··· 2 1 1 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 kernel S3×C3×C12 C32×Dic3 C32×C12 S3×C3×C6 S3×C12 S3×C32 C3×Dic3 C3×C12 S3×C6 C3×S3 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 8 4 8 8 8 32 1 1 8 2 8 16

Matrix representation of S3×C3×C12 in GL3(𝔽13) generated by

 1 0 0 0 3 0 0 0 3
,
 3 0 0 0 8 0 0 0 8
,
 1 0 0 0 3 3 0 0 9
,
 1 0 0 0 1 0 0 2 12
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[3,0,0,0,8,0,0,0,8],[1,0,0,0,3,0,0,3,9],[1,0,0,0,1,2,0,0,12] >;

S3×C3×C12 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{12}
% in TeX

G:=Group("S3xC3xC12");
// GroupNames label

G:=SmallGroup(216,136);
// by ID

G=gap.SmallGroup(216,136);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,223,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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