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## G = C3×D4×C3⋊S3order 432 = 24·33

### Direct product of C3, D4 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D4×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C2×C6×C3⋊S3 — C3×D4×C3⋊S3
 Lower central C32 — C3×C6 — C3×D4×C3⋊S3
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4×C3⋊S3
G = < a,b,c,d,e,f | a3=b4=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1380 in 388 conjugacy classes, 98 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, 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⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, S3×D4, C6×D4, C3×C3⋊S3, C3×C3⋊S3, C32×C6, C32×C6, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, D4×C32, D4×C32, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C3×S3×D4, D4×C3⋊S3, C12×C3⋊S3, C3×C12⋊S3, C3×C327D4, D4×C33, C2×C6×C3⋊S3, C3×D4×C3⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, S3×D4, C6×D4, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×S3×D4, D4×C3⋊S3, C2×C6×C3⋊S3, C3×D4×C3⋊S3

Smallest permutation representation of C3×D4×C3⋊S3
On 72 points
Generators in S72
(1 29 7)(2 30 8)(3 31 5)(4 32 6)(9 55 61)(10 56 62)(11 53 63)(12 54 64)(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 72 49)(18 69 50)(19 70 51)(20 71 52)(21 59 34)(22 60 35)(23 57 36)(24 58 33)(25 45 66)(26 46 67)(27 47 68)(28 48 65)
(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 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 42)(43 44)(45 48)(46 47)(49 50)(51 52)(53 56)(54 55)(57 58)(59 60)(61 64)(62 63)(65 66)(67 68)(69 72)(70 71)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 55 61)(10 56 62)(11 53 63)(12 54 64)(13 41 37)(14 42 38)(15 43 39)(16 44 40)(17 72 49)(18 69 50)(19 70 51)(20 71 52)(21 34 59)(22 35 60)(23 36 57)(24 33 58)(25 45 66)(26 46 67)(27 47 68)(28 48 65)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 35 42)(6 36 43)(7 33 44)(8 34 41)(9 47 50)(10 48 51)(11 45 52)(12 46 49)(17 54 67)(18 55 68)(19 56 65)(20 53 66)(25 71 63)(26 72 64)(27 69 61)(28 70 62)(29 58 40)(30 59 37)(31 60 38)(32 57 39)
(1 11)(2 12)(3 9)(4 10)(5 61)(6 62)(7 63)(8 64)(13 46)(14 47)(15 48)(16 45)(17 59)(18 60)(19 57)(20 58)(21 49)(22 50)(23 51)(24 52)(25 44)(26 41)(27 42)(28 43)(29 53)(30 54)(31 55)(32 56)(33 71)(34 72)(35 69)(36 70)(37 67)(38 68)(39 65)(40 66)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6R 6S ··· 6AP 6AQ 6AR 6AS 6AT 6AU 6AV 6AW 6AX 12A 12B 12C ··· 12N 12O 12P 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 6 6 6 6 12 12 12 ··· 12 12 12 size 1 1 2 2 9 9 18 18 1 1 2 ··· 2 2 18 1 1 2 ··· 2 4 ··· 4 9 9 9 9 18 18 18 18 2 2 4 ··· 4 18 18

90 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 C3×D4×C3⋊S3 C12×C3⋊S3 C3×C12⋊S3 C3×C32⋊7D4 D4×C33 C2×C6×C3⋊S3 D4×C3⋊S3 C4×C3⋊S3 C12⋊S3 C32⋊7D4 D4×C32 C22×C3⋊S3 D4×C32 C3×C3⋊S3 C3×C12 C62 C3×D4 C3⋊S3 C12 C2×C6 C32 C3 # reps 1 1 1 2 1 2 2 2 2 4 2 4 4 2 4 8 8 4 8 16 4 8

Matrix representation of C3×D4×C3⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 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
,
 12 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 0 12 11 0 0 0 0 1 1
,
 12 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 0 12 11 0 0 0 0 0 1
,
 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 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 3 0 0 0 0 0 3 9 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 1 2 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12

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

C3×D4×C3⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_4\times C_3\rtimes S_3
% in TeX

G:=Group("C3xD4xC3:S3");
// GroupNames label

G:=SmallGroup(432,714);
// by ID

G=gap.SmallGroup(432,714);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,4037,14118]);
// Polycyclic

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

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