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G = Dic3×C3×C6order 216 = 23·33

Direct product of C3×C6 and Dic3

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

Aliases: Dic3×C3×C6, C6.4C62, C62.17S3, C62.13C6, C6⋊(C3×C12), C32(C6×C12), (C3×C6)⋊4C12, C6.38(S3×C6), (C32×C6)⋊3C4, C3313(C2×C4), (C3×C6).70D6, C329(C2×C12), (C3×C62).1C2, C22.(S3×C32), (C32×C6).19C22, C2.2(S3×C3×C6), (C2×C6).9(C3×C6), (C2×C6).22(C3×S3), (C3×C6).27(C2×C6), SmallGroup(216,138)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C3×C6
C1C3C6C3×C6C32×C6C32×Dic3 — Dic3×C3×C6
C3 — Dic3×C3×C6
C1C62

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 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×14], C6 [×12], C2×C4, C32, C32 [×4], C32 [×4], Dic3 [×2], C12 [×8], C2×C6, C2×C6 [×4], C2×C6 [×4], C3×C6, C3×C6 [×14], C3×C6 [×12], C2×Dic3, C2×C12 [×4], C33, C3×Dic3 [×8], C3×C12 [×2], C62, C62 [×4], C62 [×4], C32×C6, C32×C6 [×2], C6×Dic3 [×4], C6×C12, C32×Dic3 [×2], C3×C62, Dic3×C3×C6
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, S3, C6 [×12], C2×C4, C32, Dic3 [×2], C12 [×8], D6, C2×C6 [×4], C3×S3 [×4], C3×C6 [×3], C2×Dic3, C2×C12 [×4], C3×Dic3 [×8], C3×C12 [×2], S3×C6 [×4], C62, S3×C32, C6×Dic3 [×4], C6×C12, C32×Dic3 [×2], S3×C3×C6, Dic3×C3×C6

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

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

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

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

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

108 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6
kernelDic3×C3×C6C32×Dic3C3×C62C6×Dic3C32×C6C3×Dic3C62C3×C6C62C3×C6C3×C6C2×C6C6C6
# reps12184168321218168

Matrix representation of Dic3×C3×C6 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] >;

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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