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G = C3×S3×Dic6  order 432 = 24·33

Direct product of C3, S3 and Dic6

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×Dic6, C12.78S32, C33⋊7(C2×Q8), D6.7(S3×C6), C3⋊1(C6×Dic6), (S3×Dic3).C6, C32⋊4(C6×Q8), (S3×C12).8S3, (S3×C12).6C6, C12.42(S3×C6), (C3×Dic6)⋊4C6, (S3×C6).46D6, C32⋊2Q8⋊2C6, (S3×C32)⋊3Q8, C32⋊13(S3×Q8), (C3×C12).176D6, Dic3.5(S3×C6), C32⋊4Q8⋊10C6, (C3×Dic3).45D6, (C32×Dic6)⋊6C2, C32⋊12(C2×Dic6), (C32×C6).20C23, (C32×C12).35C22, (C32×Dic3).8C22, C3⋊2(C3×S3×Q8), C2.4(S32×C6), C4.5(C3×S32), (C3×S3)⋊(C3×Q8), C6.1(S3×C2×C6), C6.104(C2×S32), (S3×C3×C12).3C2, (C4×S3).1(C3×S3), (C3×S3×Dic3).2C2, (S3×C6).11(C2×C6), (C3×C12).50(C2×C6), (C3×C32⋊2Q8)⋊8C2, (S3×C3×C6).18C22, C3⋊Dic3.8(C2×C6), (C3×C32⋊4Q8)⋊6C2, (C3×C6).11(C22×C6), (C3×Dic3).1(C2×C6), (C3×C6).125(C22×S3), (C3×C3⋊Dic3).34C22, SmallGroup(432,642)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×S3×Dic6
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3×S3×Dic6
Lower central
C32 — C3×C6 — C3×S3×Dic6
Upper central
C1 — C6 — C12

Generators and relations for C3×S3×Dic6
 G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=d6, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 568 in 198 conjugacy classes, 72 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C2×Dic6, S3×Q8, C6×Q8, S3×C32, C32×C6, S3×Dic3, C32⋊2Q8, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C6×Dic3, C32⋊4Q8, C6×C12, Q8×C32, C32×Dic3, C32×Dic3, C3×C3⋊Dic3, C32×C12, S3×C3×C6, S3×Dic6, C6×Dic6, C3×S3×Q8, C3×S3×Dic3, C3×C32⋊2Q8, C32×Dic6, S3×C3×C12, C3×C32⋊4Q8, C3×S3×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S32, S3×C6, C2×Dic6, S3×Q8, C6×Q8, C3×Dic6, C2×S32, S3×C2×C6, C3×S32, S3×Dic6, C6×Dic6, C3×S3×Q8, S32×C6, C3×S3×Dic6

Smallest permutation representation of C3×S3×Dic6
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)

(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)

(1 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)
 
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)>;
 
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44) );
 
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17),(25,43,31,37),(26,42,32,48),(27,41,33,47),(28,40,34,46),(29,39,35,45),(30,38,36,44)]])
 

81 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K4A4B4C4D4E4F6A6B6C···6H6I6J6K6L6M6N6O6P···6U12A···12H12I···12Q12R···12AC12AD···12AI12AJ12AK12AL12AM
order1222333···3333444444666···666666666···612···1212···1212···1212···1212121212
size1133112···244426661818112···233334446···62···24···46···612···1218181818

81 irreducible representations

dim1111111111112222222222222244444444
type++++++++-+++-+-+-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3S3Q8D6D6D6C3×S3C3×S3Dic6C3×Q8S3×C6S3×C6S3×C6C3×Dic6S32S3×Q8C2×S32C3×S32S3×Dic6C3×S3×Q8S32×C6C3×S3×Dic6
kernelC3×S3×Dic6C3×S3×Dic3C3×C32⋊2Q8C32×Dic6S3×C3×C12C3×C32⋊4Q8S3×Dic6S3×Dic3C32⋊2Q8C3×Dic6S3×C12C32⋊4Q8C3×Dic6S3×C12S3×C32C3×Dic3C3×C12S3×C6Dic6C4×S3C3×S3C3×S3Dic3C12D6S3C12C32C6C4C3C3C2C1
# reps1221112442221123212244642811122224

Matrix representation of C3×S3×Dic6 ►in GL6(𝔽13)

300000
030000
009000
000900
000010
000001
,
100000
010000
0012100
0012000
000010
000001
,
100000
010000
0011200
0001200
000010
000001
,
010000
1200000
001000
000100
00001212
000010
,
1040000
430000
0012000
0001200
000010
00001212

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
 

C3×S3×Dic6 in GAP, Magma, Sage, TeX 

C_3\times S_3\times {\rm Dic}_6
 
% in TeX
 
G:=Group("C3xS3xDic6");
 
// GroupNames label
 
G:=SmallGroup(432,642);
 
// by ID
 
G=gap.SmallGroup(432,642);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=d^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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