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## G = Dic3×C3×C6order 216 = 23·33

### Direct product of C3×C6 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C3×C6
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C32×Dic3 — Dic3×C3×C6
 Lower central C3 — Dic3×C3×C6
 Upper central C1 — C62

Generators and relations for Dic3×C3×C6
G = < a,b,c,d | a3=b6=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 212 in 136 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C3×Dic3, C3×C12, C62, C62, C62, C32×C6, C32×C6, C6×Dic3, C6×C12, C32×Dic3, C3×C62, Dic3×C3×C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, S3×C32, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, Dic3×C3×C6

Smallest permutation representation of Dic3×C3×C6
On 72 points
Generators in S72
(1 63 43)(2 64 44)(3 65 45)(4 66 46)(5 61 47)(6 62 48)(7 17 33)(8 18 34)(9 13 35)(10 14 36)(11 15 31)(12 16 32)(19 70 29)(20 71 30)(21 72 25)(22 67 26)(23 68 27)(24 69 28)(37 51 60)(38 52 55)(39 53 56)(40 54 57)(41 49 58)(42 50 59)
(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 37 65 53 47 58)(2 38 66 54 48 59)(3 39 61 49 43 60)(4 40 62 50 44 55)(5 41 63 51 45 56)(6 42 64 52 46 57)(7 72 31 19 13 27)(8 67 32 20 14 28)(9 68 33 21 15 29)(10 69 34 22 16 30)(11 70 35 23 17 25)(12 71 36 24 18 26)
(1 24 53 12)(2 19 54 7)(3 20 49 8)(4 21 50 9)(5 22 51 10)(6 23 52 11)(13 66 72 59)(14 61 67 60)(15 62 68 55)(16 63 69 56)(17 64 70 57)(18 65 71 58)(25 42 35 46)(26 37 36 47)(27 38 31 48)(28 39 32 43)(29 40 33 44)(30 41 34 45)

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

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

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

Dic3×C3×C6 is a maximal subgroup of   C62.78D6  C62.79D6  C62.80D6  C62.81D6  C62.82D6  C62.93D6  S3×C6×C12

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6X 6Y ··· 6AY 12A ··· 12AF order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 1 ··· 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel Dic3×C3×C6 C32×Dic3 C3×C62 C6×Dic3 C32×C6 C3×Dic3 C62 C3×C6 C62 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 2 1 8 4 16 8 32 1 2 1 8 16 8

Matrix representation of Dic3×C3×C6 in GL3(𝔽13) generated by

 9 0 0 0 9 0 0 0 9
,
 12 0 0 0 10 0 0 0 10
,
 12 0 0 0 9 3 0 0 3
,
 5 0 0 0 12 0 0 2 1
G:=sub<GL(3,GF(13))| [9,0,0,0,9,0,0,0,9],[12,0,0,0,10,0,0,0,10],[12,0,0,0,9,0,0,3,3],[5,0,0,0,12,2,0,0,1] >;

Dic3×C3×C6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\times C_6
% in TeX

G:=Group("Dic3xC3xC6");
// GroupNames label

G:=SmallGroup(216,138);
// by ID

G=gap.SmallGroup(216,138);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,5189]);
// Polycyclic

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

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