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## G = C3×D12.C4order 288 = 25·32

### Direct product of C3 and D12.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D12.C4
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×C4○D12 — C3×D12.C4
 Lower central C3 — C6 — C3×D12.C4
 Upper central C1 — C12 — C3×M4(2)

Generators and relations for C3×D12.C4
G = < a,b,c,d | a3=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >

Subgroups: 250 in 134 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C2×C24 [×3], C3×M4(2) [×2], C3×M4(2) [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×C24 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, D12.C4, C3×C8○D4, S3×C24 [×2], C3×C8⋊S3 [×2], C6×C3⋊C8, C32×M4(2), C3×C4○D12, C3×D12.C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C8○D4, S3×C6 [×3], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, D12.C4, C3×C8○D4, S3×C2×C12, C3×D12.C4

Smallest permutation representation of C3×D12.C4
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 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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)
(1 13 10 16 7 19 4 22)(2 20 11 23 8 14 5 17)(3 15 12 18 9 21 6 24)(25 40 28 37 31 46 34 43)(26 47 29 44 32 41 35 38)(27 42 30 39 33 48 36 45)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 12Q 12R 12S 24A ··· 24H 24I ··· 24P 24Q ··· 24AB 24AC 24AD 24AE 24AF order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 24 24 24 24 size 1 1 2 6 6 1 1 2 2 2 1 1 2 6 6 1 1 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 3 3 3 3 6 6 1 1 1 1 2 ··· 2 4 4 4 6 6 6 6 2 ··· 2 3 ··· 3 4 ··· 4 6 6 6 6

90 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 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 S3 D6 D6 C3×S3 C4×S3 C4×S3 C8○D4 S3×C6 S3×C6 S3×C12 S3×C12 C3×C8○D4 D12.C4 C3×D12.C4 kernel C3×D12.C4 S3×C24 C3×C8⋊S3 C6×C3⋊C8 C32×M4(2) C3×C4○D12 D12.C4 C3×Dic6 C3×D12 C3×C3⋊D4 S3×C8 C8⋊S3 C2×C3⋊C8 C3×M4(2) C4○D12 Dic6 D12 C3⋊D4 C3×M4(2) C24 C2×C12 M4(2) C12 C2×C6 C32 C8 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 2 2 2 4 4 4 2 2 2 4 4 8 1 2 1 2 2 2 4 4 2 4 4 8 2 4

Matrix representation of C3×D12.C4 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 65 0 0 0 0 9 0 0 0 0 46 0 0 0 0 27
,
 0 9 0 0 65 0 0 0 0 0 0 51 0 0 63 0
,
 46 0 0 0 0 46 0 0 0 0 0 1 0 0 46 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,9,0,0,0,0,46,0,0,0,0,27],[0,65,0,0,9,0,0,0,0,0,0,63,0,0,51,0],[46,0,0,0,0,46,0,0,0,0,0,46,0,0,1,0] >;

C3×D12.C4 in GAP, Magma, Sage, TeX

C_3\times D_{12}.C_4
% in TeX

G:=Group("C3xD12.C4");
// GroupNames label

G:=SmallGroup(288,678);
// by ID

G=gap.SmallGroup(288,678);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,555,142,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations

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