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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 — C12 — C3×D12⋊C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4○D12 — C3×D12⋊C4
 Lower central C3 — C6 — C12 — C3×D12⋊C4
 Upper central C1 — C12 — C2×C12 — C3×M4(2)

Generators and relations for C3×D12⋊C4
G = < a,b,c,d | a3=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 250 in 98 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×5], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×3], C12 [×4], C12 [×5], D6, C2×C6 [×2], C2×C6 [×2], C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C24 [×4], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C3×Q8, C4≀C2, C3×Dic3 [×3], C3×C12 [×2], S3×C6, C62, C4×Dic3, C4×C12, C3×M4(2) [×2], C3×M4(2), C4○D12, C3×C4○D4, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D12⋊C4, C3×C4≀C2, Dic3×C12, C32×M4(2), C3×C4○D12, C3×D12⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C4≀C2, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, D12⋊C4, C3×C4≀C2, C3×D6⋊C4, 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)
(1 40 7 46)(2 45 8 39)(3 38 9 44)(4 43 10 37)(5 48 11 42)(6 41 12 47)(13 27)(14 32)(15 25)(16 30)(17 35)(18 28)(19 33)(20 26)(21 31)(22 36)(23 29)(24 34)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 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 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D4 D4 D6 C3×S3 C4×S3 C3⋊D4 C3×D4 D12 C3×D4 C4≀C2 S3×C6 S3×C12 C3×C3⋊D4 C3×D12 C3×C4≀C2 D12⋊C4 C3×D12⋊C4 kernel C3×D12⋊C4 Dic3×C12 C32×M4(2) C3×C4○D12 D12⋊C4 C3×Dic6 C3×D12 C4×Dic3 C3×M4(2) C4○D12 Dic6 D12 C3×M4(2) C3×C12 C62 C2×C12 M4(2) C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 1 2 2 2 2 2 2 4 2 4 4 4 8 2 4

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

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 65 0 0 0 0 46 44 0 0 0 27
,
 0 65 0 0 9 0 0 0 0 0 46 44 0 0 10 27
,
 0 46 0 0 27 0 0 0 0 0 27 41 0 0 0 1
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,46,0,0,0,44,27],[0,9,0,0,65,0,0,0,0,0,46,10,0,0,44,27],[0,27,0,0,46,0,0,0,0,0,27,0,0,0,41,1] >;

C3×D12⋊C4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,259);
// by ID

G=gap.SmallGroup(288,259);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,136,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
// generators/relations

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