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## G = C3×D4.D6order 288 = 25·32

### Direct product of C3 and D4.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D4.D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×Q8 — C3×D4.D6
 Lower central C3 — C6 — C12 — C3×D4.D6
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×D4.D6
G = < a,b,c,d,e | a3=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >

Subgroups: 306 in 130 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3, C6 [×2], C6 [×5], C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×7], D6, C2×C6 [×4], M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C2×C12 [×3], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8 [×4], C8.C22, C3×Dic3, C3×Dic3 [×2], C3×C12, C3×C12, S3×C6, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×M4(2), C3×SD16 [×2], C3×SD16 [×2], C3×Q16 [×2], D42S3, S3×Q8, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6 [×2], C3×Dic6, S3×C12, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, D4.D6, C3×C8.C22, C3×C8⋊S3, C3×Dic12, C3×D4.S3, C3×C3⋊Q16, C32×SD16, C3×D42S3, C3×S3×Q8, C3×D4.D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C8.C22, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, D4.D6, C3×C8.C22, C3×S3×D4, C3×D4.D6

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 12A 12B 12C ··· 12G 12H 12I 12J 12K 12L 12M 12N 12O 12P 24A ··· 24H 24I 24J order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 ··· 12 12 12 12 12 12 12 12 12 12 24 ··· 24 24 24 size 1 1 4 6 1 1 2 2 2 2 4 6 12 12 1 1 2 2 2 4 4 6 6 8 8 8 4 12 2 2 4 ··· 4 6 6 8 8 8 12 12 12 12 4 ··· 4 12 12

54 irreducible representations

 dim 1 1 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 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D6 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 S3×C6 C8.C22 S3×D4 D4.D6 C3×C8.C22 C3×S3×D4 C3×D4.D6 kernel C3×D4.D6 C3×C8⋊S3 C3×Dic12 C3×D4.S3 C3×C3⋊Q16 C32×SD16 C3×D4⋊2S3 C3×S3×Q8 D4.D6 C8⋊S3 Dic12 D4.S3 C3⋊Q16 C3×SD16 D4⋊2S3 S3×Q8 C3×SD16 C3×Dic3 S3×C6 C24 C3×D4 C3×Q8 SD16 Dic3 D6 C8 D4 Q8 C32 C6 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 1 1 2 2 2 4

Matrix representation of C3×D4.D6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 72 0 0 0 0 0 0 1 0 0 72 0
,
 0 1 0 0 1 0 0 0 0 0 0 72 0 0 72 0
,
 19 54 0 0 54 54 0 0 0 0 48 25 0 0 25 25
,
 0 0 48 25 0 0 25 25 19 54 0 0 54 54 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,0,72,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,72,0],[19,54,0,0,54,54,0,0,0,0,48,25,0,0,25,25],[0,0,19,54,0,0,54,54,48,25,0,0,25,25,0,0] >;

C3×D4.D6 in GAP, Magma, Sage, TeX

C_3\times D_4.D_6
% in TeX

G:=Group("C3xD4.D6");
// GroupNames label

G:=SmallGroup(288,686);
// by ID

G=gap.SmallGroup(288,686);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,303,1271,648,102,9414]);
// Polycyclic

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

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