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## G = S3×D4×C32order 432 = 24·33

### Direct product of C32, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×D4×C32
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C62 — S3×D4×C32
 Lower central C3 — C6 — S3×D4×C32
 Upper central C1 — C3×C6 — D4×C32

Generators and relations for S3×D4×C32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 880 in 388 conjugacy classes, 150 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C2×C4, D4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, S3×C6, C62, C62, S3×D4, C6×D4, S3×C32, S3×C32, C32×C6, C32×C6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D4×C32, D4×C32, D4×C32, S3×C2×C6, C2×C62, C32×Dic3, C32×C12, S3×C3×C6, S3×C3×C6, S3×C3×C6, C3×C62, C3×S3×D4, D4×C3×C6, S3×C3×C12, C32×D12, C32×C3⋊D4, D4×C33, S3×C62, S3×D4×C32
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C32, D6, C2×C6, C2×D4, C3×S3, C3×C6, C3×D4, C22×S3, C22×C6, S3×C6, C62, S3×D4, C6×D4, S3×C32, D4×C32, S3×C2×C6, C2×C62, S3×C3×C6, C3×S3×D4, D4×C3×C6, S3×C62, S3×D4×C32

Smallest permutation representation of S3×D4×C32
On 72 points
Generators in S72
(1 59 7)(2 60 8)(3 57 5)(4 58 6)(9 30 27)(10 31 28)(11 32 25)(12 29 26)(13 39 34)(14 40 35)(15 37 36)(16 38 33)(17 72 54)(18 69 55)(19 70 56)(20 71 53)(21 46 41)(22 47 42)(23 48 43)(24 45 44)(49 67 63)(50 68 64)(51 65 61)(52 66 62)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 67)(18 23 68)(19 24 65)(20 21 66)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 52 53)(42 49 54)(43 50 55)(44 51 56)(45 61 70)(46 62 71)(47 63 72)(48 64 69)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 67 22)(18 68 23)(19 65 24)(20 66 21)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 53 52)(42 54 49)(43 55 50)(44 56 51)(45 70 61)(46 71 62)(47 72 63)(48 69 64)
(1 41)(2 42)(3 43)(4 44)(5 48)(6 45)(7 46)(8 47)(9 63)(10 64)(11 61)(12 62)(13 56)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 59)(22 60)(23 57)(24 58)(25 65)(26 66)(27 67)(28 68)(29 52)(30 49)(31 50)(32 51)(33 69)(34 70)(35 71)(36 72)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 26)(27 28)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 46)(47 48)(49 50)(51 52)(53 56)(54 55)(57 60)(58 59)(61 62)(63 64)(65 66)(67 68)(69 72)(70 71)

G:=sub<Sym(72)| (1,59,7)(2,60,8)(3,57,5)(4,58,6)(9,30,27)(10,31,28)(11,32,25)(12,29,26)(13,39,34)(14,40,35)(15,37,36)(16,38,33)(17,72,54)(18,69,55)(19,70,56)(20,71,53)(21,46,41)(22,47,42)(23,48,43)(24,45,44)(49,67,63)(50,68,64)(51,65,61)(52,66,62), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,22,67)(18,23,68)(19,24,65)(20,21,66)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,52,53)(42,49,54)(43,50,55)(44,51,56)(45,61,70)(46,62,71)(47,63,72)(48,64,69), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,67,22)(18,68,23)(19,65,24)(20,66,21)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,53,52)(42,54,49)(43,55,50)(44,56,51)(45,70,61)(46,71,62)(47,72,63)(48,69,64), (1,41)(2,42)(3,43)(4,44)(5,48)(6,45)(7,46)(8,47)(9,63)(10,64)(11,61)(12,62)(13,56)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,59)(22,60)(23,57)(24,58)(25,65)(26,66)(27,67)(28,68)(29,52)(30,49)(31,50)(32,51)(33,69)(34,70)(35,71)(36,72), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,46)(47,48)(49,50)(51,52)(53,56)(54,55)(57,60)(58,59)(61,62)(63,64)(65,66)(67,68)(69,72)(70,71)>;

G:=Group( (1,59,7)(2,60,8)(3,57,5)(4,58,6)(9,30,27)(10,31,28)(11,32,25)(12,29,26)(13,39,34)(14,40,35)(15,37,36)(16,38,33)(17,72,54)(18,69,55)(19,70,56)(20,71,53)(21,46,41)(22,47,42)(23,48,43)(24,45,44)(49,67,63)(50,68,64)(51,65,61)(52,66,62), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,22,67)(18,23,68)(19,24,65)(20,21,66)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,52,53)(42,49,54)(43,50,55)(44,51,56)(45,61,70)(46,62,71)(47,63,72)(48,64,69), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,67,22)(18,68,23)(19,65,24)(20,66,21)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,53,52)(42,54,49)(43,55,50)(44,56,51)(45,70,61)(46,71,62)(47,72,63)(48,69,64), (1,41)(2,42)(3,43)(4,44)(5,48)(6,45)(7,46)(8,47)(9,63)(10,64)(11,61)(12,62)(13,56)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,59)(22,60)(23,57)(24,58)(25,65)(26,66)(27,67)(28,68)(29,52)(30,49)(31,50)(32,51)(33,69)(34,70)(35,71)(36,72), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,46)(47,48)(49,50)(51,52)(53,56)(54,55)(57,60)(58,59)(61,62)(63,64)(65,66)(67,68)(69,72)(70,71) );

G=PermutationGroup([[(1,59,7),(2,60,8),(3,57,5),(4,58,6),(9,30,27),(10,31,28),(11,32,25),(12,29,26),(13,39,34),(14,40,35),(15,37,36),(16,38,33),(17,72,54),(18,69,55),(19,70,56),(20,71,53),(21,46,41),(22,47,42),(23,48,43),(24,45,44),(49,67,63),(50,68,64),(51,65,61),(52,66,62)], [(1,29,14),(2,30,15),(3,31,16),(4,32,13),(5,10,33),(6,11,34),(7,12,35),(8,9,36),(17,22,67),(18,23,68),(19,24,65),(20,21,66),(25,39,58),(26,40,59),(27,37,60),(28,38,57),(41,52,53),(42,49,54),(43,50,55),(44,51,56),(45,61,70),(46,62,71),(47,63,72),(48,64,69)], [(1,29,14),(2,30,15),(3,31,16),(4,32,13),(5,10,33),(6,11,34),(7,12,35),(8,9,36),(17,67,22),(18,68,23),(19,65,24),(20,66,21),(25,39,58),(26,40,59),(27,37,60),(28,38,57),(41,53,52),(42,54,49),(43,55,50),(44,56,51),(45,70,61),(46,71,62),(47,72,63),(48,69,64)], [(1,41),(2,42),(3,43),(4,44),(5,48),(6,45),(7,46),(8,47),(9,63),(10,64),(11,61),(12,62),(13,56),(14,53),(15,54),(16,55),(17,37),(18,38),(19,39),(20,40),(21,59),(22,60),(23,57),(24,58),(25,65),(26,66),(27,67),(28,68),(29,52),(30,49),(31,50),(32,51),(33,69),(34,70),(35,71),(36,72)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23),(25,26),(27,28),(29,32),(30,31),(33,36),(34,35),(37,38),(39,40),(41,44),(42,43),(45,46),(47,48),(49,50),(51,52),(53,56),(54,55),(57,60),(58,59),(61,62),(63,64),(65,66),(67,68),(69,72),(70,71)]])

135 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6AG 6AH ··· 6AW 6AX ··· 6BO 6BP ··· 6CE 12A ··· 12H 12I ··· 12Q 12R ··· 12Y order 1 2 2 2 2 2 2 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 2 2 3 3 6 6 1 ··· 1 2 ··· 2 2 6 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×D4 C3×S3×D4 kernel S3×D4×C32 S3×C3×C12 C32×D12 C32×C3⋊D4 D4×C33 S3×C62 C3×S3×D4 S3×C12 C3×D12 C3×C3⋊D4 D4×C32 S3×C2×C6 D4×C32 S3×C32 C3×C12 C62 C3×D4 C3×S3 C12 C2×C6 C32 C3 # reps 1 1 1 2 1 2 8 8 8 16 8 16 1 2 1 2 8 16 8 16 1 8

Matrix representation of S3×D4×C32 in GL6(𝔽13)

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

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

S3×D4×C32 in GAP, Magma, Sage, TeX

S_3\times D_4\times C_3^2
% in TeX

G:=Group("S3xD4xC3^2");
// GroupNames label

G:=SmallGroup(432,704);
// by ID

G=gap.SmallGroup(432,704);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,807,14118]);
// Polycyclic

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

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