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G = C3×D6.D6  order 432 = 24·33

Direct product of C3 and D6.D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6.D6, (S3×C12)⋊1C6, (S3×C12)⋊9S3, C12.105S32, D6.5(S3×C6), D6⋊S3⋊6C6, C3⋊D12⋊7C6, C12.48(S3×C6), (S3×C6).37D6, C32⋊2Q8⋊6C6, (C3×C12).183D6, C33⋊10(C4○D4), Dic3.6(S3×C6), (C3×Dic3).46D6, C32⋊21(C4○D12), (C32×C6).24C23, (C32×C12).72C22, (C32×Dic3).24C22, C2.8(S32×C6), C6.5(S3×C2×C6), (S3×C3×C12)⋊3C2, C4.17(C3×S32), (C4×C3⋊S3)⋊11C6, (C4×S3)⋊4(C3×S3), C6.108(C2×S32), C3⋊2(C3×C4○D12), (C12×C3⋊S3)⋊16C2, C32⋊6(C3×C4○D4), (S3×C6).12(C2×C6), (C3×C12).65(C2×C6), (S3×C3×C6).19C22, (C3×C3⋊D12)⋊15C2, (C3×D6⋊S3)⋊13C2, (C3×C32⋊2Q8)⋊13C2, (C6×C3⋊S3).46C22, C3⋊Dic3.18(C2×C6), (C3×C6).15(C22×C6), (C3×Dic3).6(C2×C6), (C3×C6).129(C22×S3), (C3×C3⋊Dic3).55C22, (C2×C3⋊S3).18(C2×C6), SmallGroup(432,646)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×D6.D6
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×D6⋊S3 — C3×D6.D6
Lower central
C32 — C3×C6 — C3×D6.D6
Upper central
C1 — C12

Generators and relations for C3×D6.D6
 G = < a,b,c,d,e | a3=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d5 >

Subgroups: 704 in 214 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, D6⋊S3, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C32×C12, S3×C3×C6, C6×C3⋊S3, D6.D6, C3×C4○D12, C3×D6⋊S3, C3×C3⋊D12, C3×C32⋊2Q8, S3×C3×C12, C12×C3⋊S3, C3×D6.D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, C4○D12, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D6.D6, C3×C4○D12, S32×C6, C3×D6.D6

Smallest permutation representation of C3×D6.D6
►On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 48)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6H6I6J6K6L···6AA6AB6AC12A12B12C12D12E···12P12Q···12V12W···12AL12AM12AN
order12222333···333344444666···66666···6661212121212···1212···1212···121212
size116618112···2444116618112···24446···6181811112···24···46···61818

90 irreducible representations

dim111111111111222222222222444444
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C4○D4C3×S3S3×C6S3×C6S3×C6C4○D12C3×C4○D4C3×C4○D12S32C2×S32C3×S32D6.D6S32×C6C3×D6.D6
kernelC3×D6.D6C3×D6⋊S3C3×C3⋊D12C3×C32⋊2Q8S3×C3×C12C12×C3⋊S3D6.D6D6⋊S3C3⋊D12C32⋊2Q8S3×C12C4×C3⋊S3S3×C12C3×Dic3C3×C12S3×C6C33C4×S3Dic3C12D6C32C32C3C12C6C4C3C2C1
# reps1121212242422222244448416112224

Matrix representation of C3×D6.D6 ►in GL6(𝔽13)

300000
030000
009000
000900
000010
000001
,
1200000
0120000
001000
000100
000001
00001212
,
010000
100000
001000
000100
000001
000010
,
500000
050000
001100
0012000
000010
000001
,
920000
1140000
0012000
001100
000010
000001

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C3×D6.D6 in GAP, Magma, Sage, TeX 

C_3\times D_6.D_6
 
% in TeX
 
G:=Group("C3xD6.D6");
 
// GroupNames label
 
G:=SmallGroup(432,646);
 
// by ID
 
G=gap.SmallGroup(432,646);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,142,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=e^2=b^3,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,e*c*e^-1=b^3*c,e*d*e^-1=d^5>;
 
// generators/relations
 

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