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## G = C4×C3⋊D12order 288 = 25·32

### Direct product of C4 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4×C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C4×C3⋊D12
 Lower central C32 — C3×C6 — C4×C3⋊D12
 Upper central C1 — C2×C4

Generators and relations for C4×C3⋊D12
G = < a,b,c,d | a4=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 818 in 215 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×10], C6 [×6], C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3 [×6], C12 [×4], C12 [×6], D6 [×2], D6 [×16], C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×10], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×S3 [×3], C22×C6, C4×D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C6.D4, C4×C12, S3×C2×C4, S3×C2×C4 [×3], C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12 [×4], S3×C12 [×2], C6×Dic3 [×3], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×D12, C4×C3⋊D4, D6⋊Dic3, C6.D12, Dic3⋊Dic3, Dic3×C12, C2×C3⋊D12, S3×C2×C12, C2×C4×C3⋊S3, C4×C3⋊D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C2×D12, C4○D12 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4×D12, C4×C3⋊D4, D6.D6, C4×S32, C2×C3⋊D12, C4×C3⋊D12

Smallest permutation representation of C4×C3⋊D12
On 48 points
Generators in S48
(1 29 24 48)(2 30 13 37)(3 31 14 38)(4 32 15 39)(5 33 16 40)(6 34 17 41)(7 35 18 42)(8 36 19 43)(9 25 20 44)(10 26 21 45)(11 27 22 46)(12 28 23 47)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 24)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 45 41)(38 42 46)(39 47 43)(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 3)(4 12)(5 11)(6 10)(7 9)(14 24)(15 23)(16 22)(17 21)(18 20)(25 35)(26 34)(27 33)(28 32)(29 31)(38 48)(39 47)(40 46)(41 45)(42 44)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E ··· 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L 12M ··· 12X order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 6 6 18 18 2 2 4 1 1 1 1 6 ··· 6 18 18 2 ··· 2 4 4 4 6 6 6 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 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 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 D6 D6 C4○D4 C4×S3 D12 C3⋊D4 C4×S3 C4○D12 S32 C3⋊D12 C2×S32 D6.D6 C4×S32 kernel C4×C3⋊D12 D6⋊Dic3 C6.D12 Dic3⋊Dic3 Dic3×C12 C2×C3⋊D12 S3×C2×C12 C2×C4×C3⋊S3 C3⋊D12 C4×Dic3 S3×C2×C4 C3×C12 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 C12 C12 D6 C6 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 3 2 1 2 4 4 4 4 8 1 2 1 2 2

Matrix representation of C4×C3⋊D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 1 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 12 12 0 0 0 0 1 0
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C4×C3⋊D12 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC3:D12");
// GroupNames label

G:=SmallGroup(288,551);
// by ID

G=gap.SmallGroup(288,551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^12=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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