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## G = C3⋊S3×C3⋊D4order 432 = 24·33

### Direct product of C3⋊S3 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C3⋊S3×C3⋊D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — C3⋊S3×C3⋊D4
 Lower central C33 — C32×C6 — C3⋊S3×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C3⋊S3×C3⋊D4
G = < a,b,c,d,e,f | a3=b3=c2=d3=e4=f2=1, ab=ba, cac=a-1, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 2456 in 388 conjugacy classes, 72 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×D4, C2×C3⋊D4, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×S32, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C32×Dic3, C335C4, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C2×C33⋊C2, C3×C62, S3×C3⋊D4, D4×C3⋊S3, Dic3×C3⋊S3, C336D4, C338D4, C32×C3⋊D4, C3315D4, C2×S3×C3⋊S3, C2×C6×C3⋊S3, C3⋊S3×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, S32, C2×C3⋊S3, S3×D4, C2×C3⋊D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×C3⋊D4, D4×C3⋊S3, C2×S3×C3⋊S3, C3⋊S3×C3⋊D4

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6G 6H ··· 6W 6X 6Y 6Z 6AA 6AB 6AC 6AD 6AE 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 6 12 12 12 12 size 1 1 2 6 9 9 18 54 2 ··· 2 4 4 4 4 6 54 2 ··· 2 4 ··· 4 12 12 12 12 18 18 18 18 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 S3×C3⋊D4 kernel C3⋊S3×C3⋊D4 Dic3×C3⋊S3 C33⋊6D4 C33⋊8D4 C32×C3⋊D4 C33⋊15D4 C2×S3×C3⋊S3 C2×C6×C3⋊S3 C3×C3⋊D4 C22×C3⋊S3 C3×C3⋊S3 C3×Dic3 S3×C6 C2×C3⋊S3 C62 C3⋊S3 C2×C6 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 2 4 4 2 5 4 4 4 4 8

Matrix representation of C3⋊S3×C3⋊D4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 11 3 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 9 0 0 0 0 0 0 7 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0
,
 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,7,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3⋊S3×C3⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C3:S3xC3:D4");
// GroupNames label

G:=SmallGroup(432,685);
// by ID

G=gap.SmallGroup(432,685);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,571,2028,14118]);
// Polycyclic

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

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