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## G = C3×C12⋊3D4order 288 = 25·32

### Direct product of C3 and C12⋊3D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C12⋊3D4
 Lower central C3 — C2×C6 — C3×C12⋊3D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C123D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 650 in 243 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×2], C6 [×2], C6 [×4], C6 [×13], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×6], C2×C6 [×2], C2×C6 [×27], C42, C2×D4, C2×D4 [×5], C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×3], C3×D4 [×18], C22×S3 [×2], C22×C6 [×4], C22×C6 [×4], C41D4, C3×Dic3 [×4], C3×C12 [×2], S3×C6 [×6], C62, C62 [×6], C4×Dic3, C4×C12, C2×D12, C2×C3⋊D4 [×4], C6×D4 [×2], C6×D4 [×6], C3×D12 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×8], C6×C12, D4×C32 [×2], S3×C2×C6 [×2], C2×C62 [×2], C123D4, C3×C41D4, Dic3×C12, C6×D12, C6×C3⋊D4 [×4], D4×C3×C6, C3×C123D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×6], C23, D6 [×3], C2×C6 [×7], C2×D4 [×3], C3×S3, C3⋊D4 [×2], C3×D4 [×6], C22×S3, C22×C6, C41D4, S3×C6 [×3], S3×D4 [×2], C2×C3⋊D4, C6×D4 [×3], C3×C3⋊D4 [×2], S3×C2×C6, C123D4, C3×C41D4, C3×S3×D4 [×2], C6×C3⋊D4, C3×C123D4

Smallest permutation representation of C3×C123D4
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(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 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 26 47 17)(2 31 48 22)(3 36 37 15)(4 29 38 20)(5 34 39 13)(6 27 40 18)(7 32 41 23)(8 25 42 16)(9 30 43 21)(10 35 44 14)(11 28 45 19)(12 33 46 24)
(1 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 6P ··· 6AE 6AF 6AG 6AH 6AI 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R order 1 2 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 6 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 1 1 4 4 12 12 1 1 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 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 C3 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C3⋊D4 S3×D4 C3×S3×D4 kernel C3×C12⋊3D4 Dic3×C12 C6×D12 C6×C3⋊D4 D4×C3×C6 C12⋊3D4 C4×Dic3 C2×D12 C2×C3⋊D4 C6×D4 C6×D4 C3×Dic3 C3×C12 C2×C12 C22×C6 C2×D4 Dic3 C12 C12 C2×C4 C23 C4 C6 C2 # reps 1 1 1 4 1 2 2 2 8 2 1 4 2 1 2 2 8 4 4 2 4 8 2 4

Matrix representation of C3×C123D4 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 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
,
 10 0 0 0 0 0 0 4 0 0 0 0 0 0 0 12 0 0 0 0 1 0 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;

C3×C123D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes_3D_4
% in TeX

G:=Group("C3xC12:3D4");
// GroupNames label

G:=SmallGroup(288,711);
// by ID

G=gap.SmallGroup(288,711);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,555,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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