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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 — C62 — C3×C62 — S3×C62 — C3×D6⋊Dic3
 Lower central C32 — C3×C6 — C3×D6⋊Dic3
 Upper central C1 — C2×C6

Generators and relations for C3×D6⋊Dic3
G = < a,b,c,d,e | a3=b6=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 640 in 210 conjugacy classes, 60 normal (52 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C62, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C32, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, S3×C2×C6, C2×C62, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, S3×C3×C6, C3×C62, D6⋊Dic3, C3×D6⋊C4, C3×C6.D4, Dic3×C3×C6, C6×C3⋊Dic3, S3×C62, C3×D6⋊Dic3
Quotients:

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 6AH ··· 6AW 12A ··· 12P 12Q 12R 12S 12T order 1 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 1 1 2 ··· 2 4 4 4 6 6 18 18 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + + - - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 S3 D4 Dic3 D6 C3×S3 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 C3×Dic3 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 S32 S3×Dic3 D6⋊S3 C3⋊D12 C3×S32 C3×S3×Dic3 C3×D6⋊S3 C3×C3⋊D12 kernel C3×D6⋊Dic3 Dic3×C3×C6 C6×C3⋊Dic3 S3×C62 D6⋊Dic3 S3×C3×C6 C6×Dic3 C2×C3⋊Dic3 S3×C2×C6 S3×C6 C6×Dic3 S3×C2×C6 C32×C6 S3×C6 C62 C2×Dic3 C22×S3 C3×C6 C3×C6 C3×C6 C3×C6 D6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 2 6 4 4 4 4 4 12 1 1 1 1 2 2 2 2

Matrix representation of C3×D6⋊Dic3 in GL8(𝔽13)

 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 11 6 0 0 0 0 0 0 6 2 0 0 0 0 0 0 0 0 11 6 0 0 0 0 0 0 6 2 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,6,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,11,6,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3×D6⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,426);
// by ID

G=gap.SmallGroup(432,426);
# by ID

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

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

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