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## G = C12×C32⋊C4order 432 = 24·33

### Direct product of C12 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C12×C32⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C6×C3⋊S3 — C6×C32⋊C4 — C12×C32⋊C4
 Lower central C32 — C12×C32⋊C4
 Upper central C1 — C12

Generators and relations for C12×C32⋊C4
G = < a,b,c,d | a12=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 444 in 104 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C42, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, C4×C12, C3×C3⋊S3, C32×C6, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C3×C3⋊Dic3, C32×C12, C3×C32⋊C4, C6×C3⋊S3, C4×C32⋊C4, C12×C3⋊S3, C6×C32⋊C4, C12×C32⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C2×C12, C32⋊C4, C4×C12, C2×C32⋊C4, C3×C32⋊C4, C4×C32⋊C4, C6×C32⋊C4, C12×C32⋊C4

Smallest permutation representation of C12×C32⋊C4
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 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)
(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 33 20 37)(2 34 21 38)(3 35 22 39)(4 36 23 40)(5 25 24 41)(6 26 13 42)(7 27 14 43)(8 28 15 44)(9 29 16 45)(10 30 17 46)(11 31 18 47)(12 32 19 48)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C ··· 4L 6A 6B 6C ··· 6H 6I 6J 6K 6L 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12AJ order 1 2 2 2 3 3 3 ··· 3 4 4 4 ··· 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 9 9 1 1 4 ··· 4 1 1 9 ··· 9 1 1 4 ··· 4 9 9 9 9 1 1 1 1 4 ··· 4 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C4 C6 C6 C12 C12 C12 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C4×C32⋊C4 C6×C32⋊C4 C12×C32⋊C4 kernel C12×C32⋊C4 C12×C3⋊S3 C6×C32⋊C4 C4×C32⋊C4 C3×C3⋊Dic3 C32×C12 C3×C32⋊C4 C4×C3⋊S3 C2×C32⋊C4 C3⋊Dic3 C3×C12 C32⋊C4 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 8 2 4 4 4 16 2 2 4 4 4 8

Matrix representation of C12×C32⋊C4 in GL4(𝔽13) generated by

 7 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 9 0 0 0 0 3 0 0 0 0 9 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 9
,
 0 0 5 0 0 0 0 5 0 5 0 0 5 0 0 0
G:=sub<GL(4,GF(13))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[9,0,0,0,0,3,0,0,0,0,9,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[0,0,0,5,0,0,5,0,5,0,0,0,0,5,0,0] >;

C12×C32⋊C4 in GAP, Magma, Sage, TeX

C_{12}\times C_3^2\rtimes C_4
% in TeX

G:=Group("C12xC3^2:C4");
// GroupNames label

G:=SmallGroup(432,630);
// by ID

G=gap.SmallGroup(432,630);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,176,14117,362,18822,1203]);
// Polycyclic

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

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