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G = Dic3xC3xC6order 216 = 23·33

Direct product of C3xC6 and Dic3

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

Aliases: Dic3xC3xC6, C6.4C62, C62.17S3, C62.13C6, C6:(C3xC12), C3:2(C6xC12), (C3xC6):4C12, C6.38(S3xC6), (C32xC6):3C4, C33:13(C2xC4), (C3xC6).70D6, C32:9(C2xC12), (C3xC62).1C2, C22.(S3xC32), (C32xC6).19C22, C2.2(S3xC3xC6), (C2xC6).9(C3xC6), (C2xC6).22(C3xS3), (C3xC6).27(C2xC6), SmallGroup(216,138)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3xC3xC6
C1C3C6C3xC6C32xC6C32xDic3 — Dic3xC3xC6
C3 — Dic3xC3xC6
C1C62

Generators and relations for Dic3xC3xC6
 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, C2xC4, C32, C32, C32, Dic3, C12, C2xC6, C2xC6, C2xC6, C3xC6, C3xC6, C3xC6, C2xDic3, C2xC12, C33, C3xDic3, C3xC12, C62, C62, C62, C32xC6, C32xC6, C6xDic3, C6xC12, C32xDic3, C3xC62, Dic3xC3xC6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C32, Dic3, C12, D6, C2xC6, C3xS3, C3xC6, C2xDic3, C2xC12, C3xDic3, C3xC12, S3xC6, C62, S3xC32, C6xDic3, C6xC12, C32xDic3, S3xC3xC6, Dic3xC3xC6

Smallest permutation representation of Dic3xC3xC6
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)]])

Dic3xC3xC6 is a maximal subgroup of   C62.78D6  C62.79D6  C62.80D6  C62.81D6  C62.82D6  C62.93D6  S3xC6xC12

108 conjugacy classes

class 1 2A2B2C3A···3H3I···3Q4A4B4C4D6A···6X6Y···6AY12A···12AF
order12223···33···344446···66···612···12
size11111···12···233331···12···23···3

108 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3xS3C3xDic3S3xC6
kernelDic3xC3xC6C32xDic3C3xC62C6xDic3C32xC6C3xDic3C62C3xC6C62C3xC6C3xC6C2xC6C6C6
# reps12184168321218168

Matrix representation of Dic3xC3xC6 in GL3(F13) generated by

900
090
009
,
1200
0100
0010
,
1200
093
003
,
500
0120
021
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] >;

Dic3xC3xC6 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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