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## G = C12×C3⋊D4order 288 = 25·32

### Direct product of C12 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×C3⋊D4 — C12×C3⋊D4
 Lower central C3 — C6 — C12×C3⋊D4
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C12×C3⋊D4
G = < a,b,c,d | a12=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 442 in 215 conjugacy classes, 90 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×6], C6 [×13], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×2], Dic3 [×2], C12 [×4], C12 [×10], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×15], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×15], C3×D4 [×4], C22×S3, C22×C6 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×2], S3×C6 [×2], C62, C62 [×2], C62 [×2], C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×C3⋊D4, C22×C12 [×2], C22×C12 [×2], C6×D4, S3×C12 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×4], C6×C12 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C4×C3⋊D4, D4×C12, Dic3×C12, C3×Dic3⋊C4, C3×D6⋊C4, C3×C6.D4, S3×C2×C12, C6×C3⋊D4, C2×C6×C12, C12×C3⋊D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C3×S3, C4×S3 [×2], C3⋊D4 [×2], C2×C12 [×6], C3×D4 [×2], C22×S3, C22×C6, C4×D4, S3×C6 [×3], S3×C2×C4, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, C3×C4○D4, S3×C12 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C4×C3⋊D4, D4×C12, S3×C2×C12, C3×C4○D12, C6×C3⋊D4, C12×C3⋊D4

Smallest permutation representation of C12×C3⋊D4
On 48 points
Generators in S48
(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 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 13 32 45)(2 14 33 46)(3 15 34 47)(4 16 35 48)(5 17 36 37)(6 18 25 38)(7 19 26 39)(8 20 27 40)(9 21 28 41)(10 22 29 42)(11 23 30 43)(12 24 31 44)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A ··· 6F 6G ··· 6AE 6AF 6AG 6AH 6AI 12A ··· 12H 12I ··· 12AJ 12AK ··· 12AV order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 6 6 1 1 2 2 2 1 1 1 1 2 2 6 ··· 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 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 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C6 C6 C12 S3 D4 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 C4×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C3⋊D4 S3×C12 C3×C4○D12 kernel C12×C3⋊D4 Dic3×C12 C3×Dic3⋊C4 C3×D6⋊C4 C3×C6.D4 S3×C2×C12 C6×C3⋊D4 C2×C6×C12 C4×C3⋊D4 C3×C3⋊D4 C4×Dic3 Dic3⋊C4 D6⋊C4 C6.D4 S3×C2×C4 C2×C3⋊D4 C22×C12 C3⋊D4 C22×C12 C3×C12 C2×C12 C22×C6 C3×C6 C22×C4 C12 C12 C2×C6 C2×C4 C23 C6 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 2 8 2 2 2 2 2 2 2 16 1 2 2 1 2 2 4 4 4 4 2 4 4 8 8 8

Matrix representation of C12×C3⋊D4 in GL3(𝔽13) generated by

 5 0 0 0 7 0 0 0 7
,
 1 0 0 0 3 0 0 9 9
,
 1 0 0 0 1 5 0 10 12
,
 1 0 0 0 12 8 0 0 1
G:=sub<GL(3,GF(13))| [5,0,0,0,7,0,0,0,7],[1,0,0,0,3,9,0,0,9],[1,0,0,0,1,10,0,5,12],[1,0,0,0,12,0,0,8,1] >;

C12×C3⋊D4 in GAP, Magma, Sage, TeX

C_{12}\times C_3\rtimes D_4
% in TeX

G:=Group("C12xC3:D4");
// GroupNames label

G:=SmallGroup(288,699);
// by ID

G=gap.SmallGroup(288,699);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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