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## G = C3×Dic3.D4order 288 = 25·32

### Direct product of C3 and Dic3.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×Dic3.D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — C3×Dic3.D4
 Lower central C3 — C2×C6 — C3×Dic3.D4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×Dic3.D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede=b3d-1 >

Subgroups: 354 in 165 conjugacy classes, 70 normal (58 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×9], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×3], C12 [×11], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×4], C2×C12 [×8], C3×Q8 [×2], C22×C6 [×2], C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×3], C3×C12 [×2], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C6×Dic3 [×2], C6×C12 [×2], C2×C62, Dic3.D4, C3×C22⋊Q8, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C6.D4, C32×C22⋊C4, C6×Dic6, Dic3×C2×C6, C3×Dic3.D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8, C4○D4, C3×S3, Dic6 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, S3×C6 [×3], C2×Dic6, S3×D4, D42S3, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6 [×2], S3×C2×C6, Dic3.D4, C3×C22⋊Q8, C6×Dic6, C3×S3×D4, C3×D42S3, C3×Dic3.D4

Smallest permutation representation of C3×Dic3.D4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(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 37 4 40)(2 42 5 39)(3 41 6 38)(7 22 10 19)(8 21 11 24)(9 20 12 23)(13 33 16 36)(14 32 17 35)(15 31 18 34)(25 46 28 43)(26 45 29 48)(27 44 30 47)
(1 22 14 27)(2 23 15 28)(3 24 16 29)(4 19 17 30)(5 20 18 25)(6 21 13 26)(7 32 47 37)(8 33 48 38)(9 34 43 39)(10 35 44 40)(11 36 45 41)(12 31 46 42)
(1 4)(2 5)(3 6)(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(13 16)(14 17)(15 18)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6S 6T ··· 6Y 12A ··· 12P 12Q ··· 12X 12Y 12Z 12AA 12AB order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 1 1 2 2 2 4 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 6 ··· 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D4 Q8 D6 D6 C4○D4 C3×S3 C3×D4 Dic6 C3×Q8 S3×C6 S3×C6 C3×C4○D4 C3×Dic6 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×Dic3.D4 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×C6.D4 C32×C22⋊C4 C6×Dic6 Dic3×C2×C6 Dic3.D4 Dic3⋊C4 C4⋊Dic3 C6.D4 C3×C22⋊C4 C2×Dic6 C22×Dic3 C3×C22⋊C4 C3×Dic3 C62 C2×C12 C22×C6 C3×C6 C22⋊C4 Dic3 C2×C6 C2×C6 C2×C4 C23 C6 C22 C6 C6 C2 C2 # reps 1 2 1 1 1 1 1 2 4 2 2 2 2 2 1 2 2 2 1 2 2 4 4 4 4 2 4 8 1 1 2 2

Matrix representation of C3×Dic3.D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 9 4
,
 12 0 0 0 0 12 0 0 0 0 1 8 0 0 3 12
,
 0 12 0 0 1 0 0 0 0 0 5 0 0 0 2 8
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,10,9,0,0,0,4],[12,0,0,0,0,12,0,0,0,0,1,3,0,0,8,12],[0,1,0,0,12,0,0,0,0,0,5,2,0,0,0,8],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;

C3×Dic3.D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3.D_4
% in TeX

G:=Group("C3xDic3.D4");
// GroupNames label

G:=SmallGroup(288,649);
// by ID

G=gap.SmallGroup(288,649);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,590,555,142,9414]);
// Polycyclic

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

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