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## G = C3×D6.Dic3order 432 = 24·33

### Direct product of C3 and D6.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D6.Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — S3×C3×C12 — C3×D6.Dic3
 Lower central C32 — C3×C6 — C3×D6.Dic3
 Upper central C1 — C12

Generators and relations for C3×D6.Dic3
G = < a,b,c,d,e | a3=b6=c2=1, d6=b3, e2=b3d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d5 >

Subgroups: 304 in 118 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C4.Dic3, C3×M4(2), S3×C32, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C324C8, C3×C24, S3×C12, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, D6.Dic3, C3×C8⋊S3, C3×C4.Dic3, C32×C3⋊C8, C3×C324C8, S3×C3×C12, C3×D6.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S32, S3×C6, C8⋊S3, C4.Dic3, C3×M4(2), S3×Dic3, S3×C12, C6×Dic3, C3×S32, D6.Dic3, C3×C8⋊S3, C3×C4.Dic3, C3×S3×Dic3, C3×D6.Dic3

Smallest permutation representation of C3×D6.Dic3
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(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 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 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 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 22 10 19 7 16 4 13)(2 15 11 24 8 21 5 18)(3 20 12 17 9 14 6 23)(25 46 34 43 31 40 28 37)(26 39 35 48 32 45 29 42)(27 44 36 41 33 38 30 47)

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

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

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

90 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12V 12W ··· 12AD 24A ··· 24P 24Q 24R 24S 24T order 1 2 2 3 3 3 ··· 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 24 24 24 size 1 1 6 1 1 2 ··· 2 4 4 4 1 1 6 1 1 2 ··· 2 4 4 4 6 ··· 6 6 6 18 18 1 1 1 1 2 ··· 2 4 ··· 4 6 ··· 6 6 ··· 6 18 18 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 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×S3 C4×S3 C3×Dic3 S3×C6 C3×Dic3 C8⋊S3 C4.Dic3 C3×M4(2) S3×C12 C3×C8⋊S3 C3×C4.Dic3 S32 S3×Dic3 C3×S32 D6.Dic3 C3×S3×Dic3 C3×D6.Dic3 kernel C3×D6.Dic3 C32×C3⋊C8 C3×C32⋊4C8 S3×C3×C12 D6.Dic3 C32×Dic3 S3×C3×C6 C3×C3⋊C8 C32⋊4C8 S3×C12 C3×Dic3 S3×C6 C3×C3⋊C8 S3×C12 C3×Dic3 C3×C12 S3×C6 C33 C3⋊C8 C4×S3 C3×C6 Dic3 C12 D6 C32 C32 C32 C6 C3 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 2 1 2 2 2 2 2 4 2 4 4 4 4 8 8 1 1 2 2 2 4

Matrix representation of C3×D6.Dic3 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 57 60 0 0 0 0 14 16 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 0 27 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,14,0,0,0,0,60,16,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×D6.Dic3 in GAP, Magma, Sage, TeX

C_3\times D_6.{\rm Dic}_3
% in TeX

G:=Group("C3xD6.Dic3");
// GroupNames label

G:=SmallGroup(432,416);
// by ID

G=gap.SmallGroup(432,416);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=b^3,e^2=b^3*d^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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