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## G = C3×Dic3⋊Dic3order 432 = 24·33

### Direct product of C3 and Dic3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×Dic3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — Dic3×C3×C6 — C3×Dic3⋊Dic3
 Lower central C32 — C3×C6 — C3×Dic3⋊Dic3
 Upper central C1 — C2×C6

Generators and relations for C3×Dic3⋊Dic3
G = < a,b,c,d,e | a3=b6=d6=1, c2=b3, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 448 in 162 conjugacy classes, 60 normal (52 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C32×Dic3, C3×C3⋊Dic3, C3×C62, Dic3⋊Dic3, C3×Dic3⋊C4, C3×C4⋊Dic3, Dic3×C3×C6, C6×C3⋊Dic3, C3×Dic3⋊Dic3
Quotients:

Smallest permutation representation of C3×Dic3⋊Dic3
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(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 32 4 35)(2 31 5 34)(3 36 6 33)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
(1 11 3 7 5 9)(2 12 4 8 6 10)(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 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(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)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,32,4,35)(2,31,5,34)(3,36,6,33)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,11,3,7,5,9)(2,12,4,8,6,10)(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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,32,4,35)(2,31,5,34)(3,36,6,33)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,11,3,7,5,9)(2,12,4,8,6,10)(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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(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)], [(1,32,4,35),(2,31,5,34),(3,36,6,33),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)], [(1,11,3,7,5,9),(2,12,4,8,6,10),(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)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(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 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 12A ··· 12AF 12AG 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 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 6 6 6 6 18 18 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

90 irreducible representations

 dim 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 4 4 type + + + + + - - + - + + - + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 Dic3 D6 C3×S3 Dic6 C4×S3 D12 C3⋊D4 C3×D4 C3×Q8 C3×Dic3 S3×C6 C3×Dic6 S3×C12 C3×D12 C3×C3⋊D4 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 C3×S32 C3×S3×Dic3 C3×C3⋊D12 C3×C32⋊2Q8 kernel C3×Dic3⋊Dic3 Dic3×C3×C6 C6×C3⋊Dic3 Dic3⋊Dic3 C32×Dic3 C6×Dic3 C2×C3⋊Dic3 C3×Dic3 C6×Dic3 C32×C6 C32×C6 C3×Dic3 C62 C2×Dic3 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 Dic3 C2×C6 C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 2 1 1 2 2 4 4 2 2 2 2 2 4 4 8 4 4 4 1 1 1 1 2 2 2 2

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

 3 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 3 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 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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 12 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
,
 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 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 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 12 12

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

C3×Dic3⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3:Dic3");
// GroupNames label

G:=SmallGroup(432,428);
// by ID

G=gap.SmallGroup(432,428);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,2028,14118]);
// Polycyclic

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

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