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## G = C3×Dic3.D6order 432 = 24·33

### Direct product of C3 and Dic3.D6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Dic3.D6
G = < a,b,c,d,e | a3=b6=1, c2=d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=b3d5 >

Subgroups: 608 in 198 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×Q8, C6×Q8, C3×C3⋊S3, C32×C6, C6.D6, C322Q8, C3×Dic6, C3×Dic6, S3×C12, C4×C3⋊S3, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, Dic3.D6, C3×S3×Q8, C3×C6.D6, C3×C322Q8, C32×Dic6, C12×C3⋊S3, C3×Dic3.D6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3×Q8, C22×S3, C22×C6, S32, S3×C6, S3×Q8, C6×Q8, C2×S32, S3×C2×C6, C3×S32, Dic3.D6, C3×S3×Q8, S32×C6, C3×Dic3.D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 12A 12B 12C ··· 12N 12O ··· 12V 12W ··· 12AH 12AI 12AJ order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 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 size 1 1 9 9 1 1 2 ··· 2 4 4 4 2 6 6 6 6 18 1 1 2 ··· 2 4 4 4 9 9 9 9 2 2 4 ··· 4 6 ··· 6 12 ··· 12 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + + + - + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 Q8 D6 D6 C3×S3 C3×Q8 S3×C6 S3×C6 S32 S3×Q8 C2×S32 C3×S32 Dic3.D6 C3×S3×Q8 S32×C6 C3×Dic3.D6 kernel C3×Dic3.D6 C3×C6.D6 C3×C32⋊2Q8 C32×Dic6 C12×C3⋊S3 Dic3.D6 C6.D6 C32⋊2Q8 C3×Dic6 C4×C3⋊S3 C3×Dic6 C3×C3⋊S3 C3×Dic3 C3×C12 Dic6 C3⋊S3 Dic3 C12 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 2 1 2 4 4 4 2 2 2 4 2 4 4 8 4 1 2 1 2 2 4 2 4

Matrix representation of C3×Dic3.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 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×Dic3.D6 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3.D6");
// GroupNames label

G:=SmallGroup(432,645);
// by ID

G=gap.SmallGroup(432,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,176,590,303,142,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=1,c^2=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^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=b^3*d^5>;
// generators/relations

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