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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 — S32×C6 — C3×D6⋊D6
 Lower central C32 — C3×C6 — C3×D6⋊D6
 Upper central C1 — C6 — C12

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

Subgroups: 1056 in 270 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C6×D4, S3×C32, C3×C3⋊S3, C32×C6, D6⋊S3, S3×C12, C3×D12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, D4×C32, C2×S32, S3×C2×C6, C3×C3⋊Dic3, C32×C12, C3×S32, S3×C3×C6, C6×C3⋊S3, D6⋊D6, C3×S3×D4, C3×D6⋊S3, C32×D12, C12×C3⋊S3, S32×C6, C3×D6⋊D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, S32, S3×C6, S3×D4, C6×D4, C2×S32, S3×C2×C6, C3×S32, D6⋊D6, C3×S3×D4, S32×C6, C3×D6⋊D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 6T 6U 6V 6W 6X ··· 6AI 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 6 6 6 6 6 ··· 6 12 12 12 ··· 12 12 12 size 1 1 6 6 6 6 9 9 1 1 2 ··· 2 4 4 4 2 18 1 1 2 ··· 2 4 4 4 6 ··· 6 9 9 9 9 12 ··· 12 2 2 4 ··· 4 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 D4 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S32 S3×D4 C2×S32 C3×S32 D6⋊D6 C3×S3×D4 S32×C6 C3×D6⋊D6 kernel C3×D6⋊D6 C3×D6⋊S3 C32×D12 C12×C3⋊S3 S32×C6 D6⋊D6 D6⋊S3 C3×D12 C4×C3⋊S3 C2×S32 C3×D12 C3×C3⋊S3 C3×C12 S3×C6 D12 C3⋊S3 C12 D6 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 2 2 4 4 2 4 2 2 2 4 4 4 4 8 1 2 1 2 2 4 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 12 12 0 0 0 0 1 0
,
 7 2 0 0 0 0 2 6 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 0 1
,
 11 7 0 0 0 0 7 2 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 12 12
,
 11 7 0 0 0 0 7 2 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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[7,2,0,0,0,0,2,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1],[11,7,0,0,0,0,7,2,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[11,7,0,0,0,0,7,2,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×D6⋊D6 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes D_6
% in TeX

G:=Group("C3xD6:D6");
// GroupNames label

G:=SmallGroup(432,650);
// by ID

G=gap.SmallGroup(432,650);
# by ID

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

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

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