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G = C3⋊S3×C3⋊D4  order 432 = 24·33

Direct product of C3⋊S3 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C3⋊S3×C3⋊D4, C62⋊22D6, (S3×C6)⋊5D6, C33⋊25(C2×D4), (C3×Dic3)⋊4D6, C32⋊17(S3×D4), C33⋊8D4⋊9C2, C33⋊15D4⋊4C2, C33⋊6D4⋊10C2, (C3×C62)⋊4C22, C33⋊5C4⋊6C22, (C32×C6).63C23, (C32×Dic3)⋊7C22, (C2×C6)⋊5S32, C3⋊5(D4×C3⋊S3), C6.73(C2×S32), D6⋊3(C2×C3⋊S3), C3⋊4(S3×C3⋊D4), (C2×C3⋊S3)⋊21D6, (C3×C3⋊S3)⋊11D4, (C3×C3⋊D4)⋊3S3, C22⋊4(S3×C3⋊S3), (S3×C3×C6)⋊13C22, (Dic3×C3⋊S3)⋊5C2, Dic3⋊1(C2×C3⋊S3), (C22×C3⋊S3)⋊10S3, (C6×C3⋊S3)⋊18C22, (C32×C3⋊D4)⋊7C2, C6.26(C22×C3⋊S3), C32⋊19(C2×C3⋊D4), (C3×C6).151(C22×S3), (C2×C33⋊C2)⋊7C22, (C2×S3×C3⋊S3)⋊9C2, (C2×C6×C3⋊S3)⋊5C2, (C2×C6)⋊8(C2×C3⋊S3), C2.26(C2×S3×C3⋊S3), SmallGroup(432,685)

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, C32⋊7D4, D4×C32, C2×S32, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C32×Dic3, C33⋊5C4, 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, C33⋊6D4, C33⋊8D4, C32×C3⋊D4, C33⋊15D4, 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 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6G6H···6W6X6Y6Z6AA6AB6AC6AD6AE12A12B12C12D
order122222223···33333446···66···66666666612121212
size11269918542···244446542···24···4121212121818181812121212

54 irreducible representations

dim11111111222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4
kernelC3⋊S3×C3⋊D4Dic3×C3⋊S3C33⋊6D4C33⋊8D4C32×C3⋊D4C33⋊15D4C2×S3×C3⋊S3C2×C6×C3⋊S3C3×C3⋊D4C22×C3⋊S3C3×C3⋊S3C3×Dic3S3×C6C2×C3⋊S3C62C3⋊S3C2×C6C32C6C3
# reps11111111412442544448

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

10000000
01000000
000120000
001120000
00001000
00000100
00000010
00000001
,
10000000
01000000
001210000
001200000
000011300
000012100
00000010
00000001
,
120000000
012000000
00010000
00100000
00004900
00007900
00000010
00000001
,
121000000
120000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00100000
00010000
000012000
000001200
00000001
000000120
,
012000000
120000000
001200000
000120000
000012000
000001200
00000001
00000010

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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