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## G = C3×D4○D12order 288 = 25·32

### Direct product of C3 and D4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D4○D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C3×S3×D4 — C3×D4○D12
 Lower central C3 — C6 — C3×D4○D12
 Upper central C1 — C6 — C3×C4○D4

Generators and relations for C3×D4○D12
G = < a,b,c,d,e | a3=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 834 in 352 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C2×D12, C4○D12, S3×D4, Q83S3, C6×D4, C3×C4○D4, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D4×C32, Q8×C32, S3×C2×C6, D4○D12, C3×2+ 1+4, C6×D12, C3×C4○D12, C3×S3×D4, C3×Q83S3, C32×C4○D4, C3×D4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2+ 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, D4○D12, C3×2+ 1+4, S3×C22×C6, C3×D4○D12

Smallest permutation representation of C3×D4○D12
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 16 7 22)(2 17 8 23)(3 18 9 24)(4 19 10 13)(5 20 11 14)(6 21 12 15)(25 40 31 46)(26 41 32 47)(27 42 33 48)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6K 6L ··· 6T 6U ··· 6AF 12A ··· 12N 12O ··· 12W 12X 12Y 12Z 12AA order 1 2 2 2 2 2 ··· 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 2 2 2 6 ··· 6 1 1 2 2 2 2 2 2 2 6 6 1 1 2 ··· 2 4 ··· 4 6 ··· 6 2 ··· 2 4 ··· 4 6 6 6 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 D6 C3×S3 S3×C6 S3×C6 S3×C6 2+ 1+4 D4○D12 C3×2+ 1+4 C3×D4○D12 kernel C3×D4○D12 C6×D12 C3×C4○D12 C3×S3×D4 C3×Q8⋊3S3 C32×C4○D4 D4○D12 C2×D12 C4○D12 S3×D4 Q8⋊3S3 C3×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C32 C3 C3 C1 # reps 1 3 3 6 2 1 2 6 6 12 4 2 1 3 3 1 2 6 6 2 1 2 2 4

Matrix representation of C3×D4○D12 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 12 2 0 0 12 1 0 0 0 0 12 2 0 0 12 1
,
 1 0 0 0 1 12 0 0 0 0 1 0 0 0 1 12
,
 7 0 0 0 0 7 0 0 0 0 2 0 0 0 0 2
,
 0 0 2 0 0 0 0 2 7 0 0 0 0 7 0 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[12,12,0,0,2,1,0,0,0,0,12,12,0,0,2,1],[1,1,0,0,0,12,0,0,0,0,1,1,0,0,0,12],[7,0,0,0,0,7,0,0,0,0,2,0,0,0,0,2],[0,0,7,0,0,0,0,7,2,0,0,0,0,2,0,0] >;

C3×D4○D12 in GAP, Magma, Sage, TeX

C_3\times D_4\circ D_{12}
% in TeX

G:=Group("C3xD4oD12");
// GroupNames label

G:=SmallGroup(288,999);
// by ID

G=gap.SmallGroup(288,999);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,555,1571,192,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=e^2=1,d^6=b^2,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,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations

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