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## G = C12×C3⋊S3order 216 = 23·33

### Direct product of C12 and C3⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12×C3⋊S3
G = < a,b,c,d | a12=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 304 in 120 conjugacy classes, 46 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⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C12×C3⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, C3⋊S3, C4×S3, C2×C12, S3×C6, C2×C3⋊S3, C3×C3⋊S3, S3×C12, C4×C3⋊S3, C6×C3⋊S3, C12×C3⋊S3

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A 6B 6C ··· 6N 6O 6P 6Q 6R 12A 12B 12C 12D 12E ··· 12AB 12AC 12AD 12AE 12AF order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 9 9 1 1 2 ··· 2 1 1 9 9 1 1 2 ··· 2 9 9 9 9 1 1 1 1 2 ··· 2 9 9 9 9

72 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 C12×C3⋊S3 C3×C3⋊Dic3 C32×C12 C6×C3⋊S3 C4×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 2 4 2 2 2 8 4 4 8 8 8 16

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

 4 0 0 0 0 4 0 0 0 0 7 0 0 0 0 7
,
 9 11 0 0 0 3 0 0 0 0 3 4 0 0 0 9
,
 3 2 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 5 1 0 0 2 8 0 0 0 0 1 0 0 0 8 12
G:=sub<GL(4,GF(13))| [4,0,0,0,0,4,0,0,0,0,7,0,0,0,0,7],[9,0,0,0,11,3,0,0,0,0,3,0,0,0,4,9],[3,0,0,0,2,9,0,0,0,0,1,0,0,0,0,1],[5,2,0,0,1,8,0,0,0,0,1,8,0,0,0,12] >;

C12×C3⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C12xC3:S3");
// GroupNames label

G:=SmallGroup(216,141);
// by ID

G=gap.SmallGroup(216,141);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,1444,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^3=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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