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## G = C3×C12⋊S3order 216 = 23·33

### Direct product of C3 and C12⋊S3

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 392 in 120 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, C3×D12, C12⋊S3, C32×C12, C6×C3⋊S3, C3×C12⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×C12⋊S3

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

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

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

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

C3×C12⋊S3 is a maximal subgroup of
C336D8  C338D8  C3313SD16  C3316SD16  C339D8  C3318SD16  C3×S3×D12  C12.39S32  C12.57S32  C12⋊S32  C12⋊S312S3  C123S32  C3×D4×C3⋊S3

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4 6A 6B 6C ··· 6N 6O 6P 6Q 6R 12A ··· 12Z order 1 2 2 2 3 3 3 ··· 3 4 6 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 18 18 1 1 2 ··· 2 2 1 1 2 ··· 2 18 18 18 18 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 C3×D12 kernel C3×C12⋊S3 C32×C12 C6×C3⋊S3 C12⋊S3 C3×C12 C2×C3⋊S3 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 2 2 2 4 4 1 4 8 8 2 8 16

Matrix representation of C3×C12⋊S3 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 10 10 0 0 3 7 0 0 0 0 8 0 0 0 0 5
,
 12 1 0 0 12 0 0 0 0 0 9 0 0 0 0 3
,
 10 10 0 0 7 3 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[10,3,0,0,10,7,0,0,0,0,8,0,0,0,0,5],[12,12,0,0,1,0,0,0,0,0,9,0,0,0,0,3],[10,7,0,0,10,3,0,0,0,0,0,1,0,0,1,0] >;

C3×C12⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC12:S3");
// GroupNames label

G:=SmallGroup(216,142);
// by ID

G=gap.SmallGroup(216,142);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,1444,5189]);
// 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,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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