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## G = C3×C4⋊(C32⋊C4)  order 432 = 24·33

### Direct product of C3 and C4⋊(C32⋊C4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C4⋊(C32⋊C4)
G = < a,b,c,d,e | a3=b4=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 444 in 96 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C4 [×3], C22, S3 [×4], C6, C6 [×6], C2×C4 [×3], C32, C32 [×4], Dic3 [×2], C12, C12 [×7], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3 [×2], C2×C12 [×3], C33, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C3×C12 [×4], C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, C3×C4⋊C4, C3×C3⋊S3 [×2], C32×C6, S3×C12 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C3×C3⋊Dic3, C32×C12, C3×C32⋊C4 [×2], C6×C3⋊S3, C4⋊(C32⋊C4), C12×C3⋊S3, C6×C32⋊C4 [×2], C3×C4⋊(C32⋊C4)
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, C12 [×2], C2×C6, C4⋊C4, C2×C12, C3×D4, C3×Q8, C32⋊C4, C3×C4⋊C4, C2×C32⋊C4, C3×C32⋊C4, C4⋊(C32⋊C4), C6×C32⋊C4, C3×C4⋊(C32⋊C4)

Smallest permutation representation of C3×C4⋊(C32⋊C4)
On 48 points
Generators in S48
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 38 47)(14 39 48)(15 40 45)(16 37 46)(29 43 36)(30 44 33)(31 41 34)(32 42 35)
(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)
(13 47 38)(14 48 39)(15 45 40)(16 46 37)(29 43 36)(30 44 33)(31 41 34)(32 42 35)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 47 38)(14 48 39)(15 45 40)(16 46 37)(29 43 36)(30 44 33)(31 41 34)(32 42 35)
(1 38 11 34)(2 37 12 33)(3 40 9 36)(4 39 10 35)(5 48 24 32)(6 47 21 31)(7 46 22 30)(8 45 23 29)(13 28 41 19)(14 27 42 18)(15 26 43 17)(16 25 44 20)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B ··· 4F 6A 6B 6C ··· 6H 6I 6J 6K 6L 12A 12B 12C ··· 12N 12O ··· 12X order 1 2 2 2 3 3 3 ··· 3 4 4 ··· 4 6 6 6 ··· 6 6 6 6 6 12 12 12 ··· 12 12 ··· 12 size 1 1 9 9 1 1 4 ··· 4 2 18 ··· 18 1 1 4 ··· 4 9 9 9 9 2 2 4 ··· 4 18 ··· 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C4⋊(C32⋊C4) C6×C32⋊C4 C3×C4⋊(C32⋊C4) kernel C3×C4⋊(C32⋊C4) C12×C3⋊S3 C6×C32⋊C4 C4⋊(C32⋊C4) C3×C3⋊Dic3 C32×C12 C4×C3⋊S3 C2×C32⋊C4 C3⋊Dic3 C3×C12 C3×C3⋊S3 C3×C3⋊S3 C3⋊S3 C3⋊S3 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 2 2 2 2 4 4 4 8

Matrix representation of C3×C4⋊(C32⋊C4) in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 11 0 0 0 0 1 12 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
,
 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 4 0 9 0 0 0 1 0 0 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 3 9 0 0 0 0 3 0 9 0 0 0 0 0 0 3
,
 8 0 0 0 0 0 8 5 0 0 0 0 0 0 1 0 2 0 0 0 0 0 12 1 0 0 0 1 12 0 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,4,1,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,3,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,2,12,12,12,0,0,0,1,0,0] >;

C3×C4⋊(C32⋊C4) in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes (C_3^2\rtimes C_4)
% in TeX

G:=Group("C3xC4:(C3^2:C4)");
// GroupNames label

G:=SmallGroup(432,631);
// by ID

G=gap.SmallGroup(432,631);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,176,14117,362,18822,1203]);
// Polycyclic

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

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