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## G = C3×D6⋊D4order 288 = 25·32

### Direct product of C3 and D6⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×D6⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C22×C6 — C3×D6⋊D4
 Lower central C3 — C2×C6 — C3×D6⋊D4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×D6⋊D4
G = < a,b,c,d,e | a3=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >

Subgroups: 802 in 277 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×5], C6 [×2], C6 [×4], C6 [×14], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], C32, Dic3, C12 [×7], D6 [×4], D6 [×15], C2×C6 [×2], C2×C6 [×4], C2×C6 [×28], C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, C3×S3 [×5], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×3], C3×D4 [×6], C22×S3, C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C22×C6 [×10], C22≀C2, C3×Dic3, C3×C12 [×2], S3×C6 [×4], S3×C6 [×15], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C22⋊C4 [×3], C2×D12 [×2], C2×C3⋊D4, C6×D4 [×3], S3×C23, C23×C6, C3×D12 [×4], C6×Dic3, C3×C3⋊D4 [×2], C6×C12 [×2], S3×C2×C6, S3×C2×C6 [×2], S3×C2×C6 [×6], C2×C62, D6⋊D4, C3×C22≀C2, C3×D6⋊C4 [×2], C32×C22⋊C4, C6×D12 [×2], C6×C3⋊D4, S3×C22×C6, C3×D6⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×6], C23, D6 [×3], C2×C6 [×7], C2×D4 [×3], C3×S3, D12 [×2], C3×D4 [×6], C22×S3, C22×C6, C22≀C2, S3×C6 [×3], C2×D12, S3×D4 [×2], C6×D4 [×3], C3×D12 [×2], S3×C2×C6, D6⋊D4, C3×C22≀C2, C6×D12, C3×S3×D4 [×2], C3×D6⋊D4

Smallest permutation representation of C3×D6⋊D4
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(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 33)(2 32)(3 31)(4 36)(5 35)(6 34)(7 29)(8 28)(9 27)(10 26)(11 25)(12 30)(13 41)(14 40)(15 39)(16 38)(17 37)(18 42)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 23 14 30)(2 24 15 25)(3 19 16 26)(4 20 17 27)(5 21 18 28)(6 22 13 29)(7 31 43 38)(8 32 44 39)(9 33 45 40)(10 34 46 41)(11 35 47 42)(12 36 48 37)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)(13 47)(14 46)(15 45)(16 44)(17 43)(18 48)(19 39)(20 38)(21 37)(22 42)(23 41)(24 40)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 3D 3E 4A 4B 4C 6A ··· 6F 6G ··· 6S 6T ··· 6Y 6Z ··· 6AG 6AH 6AI 12A ··· 12P 12Q 12R order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 6 12 ··· 12 12 12 size 1 1 1 1 2 2 6 6 6 6 12 1 1 2 2 2 4 4 12 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 12 12 4 ··· 4 12 12

72 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 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3×D4 D12 C3×D4 S3×C6 S3×C6 C3×D12 S3×D4 C3×S3×D4 kernel C3×D6⋊D4 C3×D6⋊C4 C32×C22⋊C4 C6×D12 C6×C3⋊D4 S3×C22×C6 D6⋊D4 D6⋊C4 C3×C22⋊C4 C2×D12 C2×C3⋊D4 S3×C23 C3×C22⋊C4 S3×C6 C62 C2×C12 C22×C6 C22⋊C4 D6 C2×C6 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 1 4 2 2 1 2 8 4 4 4 2 8 2 4

Matrix representation of C3×D6⋊D4 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 0 0 0 0 0 0 9 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 0 12
,
 0 9 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 2 7 0 0 0 0 7 11
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 2 0 0 0 0 0 12 0 0 0 0 0 0 7 11 0 0 0 0 11 6

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,3,0,0,0,0,9,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,2,7,0,0,0,0,7,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,7,11,0,0,0,0,11,6] >;

C3×D6⋊D4 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes D_4
% in TeX

G:=Group("C3xD6:D4");
// GroupNames label

G:=SmallGroup(288,653);
// by ID

G=gap.SmallGroup(288,653);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^3*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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