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## G = S32×C12order 432 = 24·33

### Direct product of C12, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C12
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — S32×C6 — S32×C12
 Lower central C32 — S32×C12
 Upper central C1 — C12

Generators and relations for S32×C12
G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 832 in 270 conjugacy classes, 92 normal (24 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, D6, C2×C6, C22×C4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C2×C4, C22×C12, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, S3×C12, S3×C12, C6×Dic3, C4×C3⋊S3, C6×C12, C2×S32, S3×C2×C6, C32×Dic3, C3×C3⋊Dic3, C32×C12, C3×S32, S3×C3×C6, C6×C3⋊S3, C4×S32, S3×C2×C12, C3×S3×Dic3, C3×C6.D6, S3×C3×C12, C12×C3⋊S3, S32×C6, S32×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S32, S3×C6, S3×C2×C4, C22×C12, S3×C12, C2×S32, S3×C2×C6, C3×S32, C4×S32, S3×C2×C12, S32×C6, S32×C12

Smallest permutation representation of S32×C12
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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)>;

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C ··· 6H 6I ··· 6P 6Q 6R 6S 6T ··· 6AE 6AF 6AG 6AH 6AI 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12X 12Y ··· 12AD 12AE ··· 12AP 12AQ 12AR 12AS 12AT order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 3 3 3 3 9 9 1 1 2 ··· 2 4 4 4 1 1 3 3 3 3 9 9 1 1 2 ··· 2 3 ··· 3 4 4 4 6 ··· 6 9 9 9 9 1 1 1 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 9 9 9 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D6 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C6 S3×C12 S32 C2×S32 C3×S32 C4×S32 S32×C6 S32×C12 kernel S32×C12 C3×S3×Dic3 C3×C6.D6 S3×C3×C12 C12×C3⋊S3 S32×C6 C4×S32 C3×S32 S3×Dic3 C6.D6 S3×C12 C4×C3⋊S3 C2×S32 S32 S3×C12 C3×Dic3 C3×C12 S3×C6 C4×S3 C3×S3 Dic3 C12 D6 S3 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 1 1 2 8 4 2 4 2 2 16 2 2 2 2 4 8 4 4 4 16 1 1 2 2 2 4

Matrix representation of S32×C12 in GL4(𝔽13) generated by

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

S32×C12 in GAP, Magma, Sage, TeX

S_3^2\times C_{12}
% in TeX

G:=Group("S3^2xC12");
// GroupNames label

G:=SmallGroup(432,648);
// by ID

G=gap.SmallGroup(432,648);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,2028,14118]);
// Polycyclic

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

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