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## G = C3×D6.D6order 432 = 24·33

### Direct product of C3 and D6.D6

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, C322Q8, 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×C322Q8, 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 2A 2B 2C 2D 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6AA 6AB 6AC 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12V 12W ··· 12AL 12AM 12AN order 1 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 12 12 size 1 1 6 6 18 1 1 2 ··· 2 4 4 4 1 1 6 6 18 1 1 2 ··· 2 4 4 4 6 ··· 6 18 18 1 1 1 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S32 C2×S32 C3×S32 D6.D6 S32×C6 C3×D6.D6 kernel C3×D6.D6 C3×D6⋊S3 C3×C3⋊D12 C3×C32⋊2Q8 S3×C3×C12 C12×C3⋊S3 D6.D6 D6⋊S3 C3⋊D12 C32⋊2Q8 S3×C12 C4×C3⋊S3 S3×C12 C3×Dic3 C3×C12 S3×C6 C33 C4×S3 Dic3 C12 D6 C32 C32 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 2 2 4 4 4 4 8 4 16 1 1 2 2 2 4

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

 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 1 0 0 0 0 12 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 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 2 0 0 0 0 11 4 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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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