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## G = C4×S3×C3⋊S3order 432 = 24·33

### Direct product of C4, S3 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C4×S3×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — C4×S3×C3⋊S3
 Lower central C33 — C4×S3×C3⋊S3
 Upper central C1 — C4

Generators and relations for C4×S3×C3⋊S3
G = < a,b,c,d,e,f | a4=b3=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: 2136 in 388 conjugacy classes, 90 normal (32 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⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, C6.D6, S3×C12, S3×C12, C4×C3⋊S3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×S32, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C335C4, C32×C12, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C4×S32, C2×C4×C3⋊S3, S3×C3⋊Dic3, Dic3×C3⋊S3, C338(C2×C4), S3×C3×C12, C12×C3⋊S3, C4×C33⋊C2, C2×S3×C3⋊S3, C4×S3×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, S32, C2×C3⋊S3, S3×C2×C4, C4×C3⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C4×S32, C2×C4×C3⋊S3, C2×S3×C3⋊S3, C4×S3×C3⋊S3

Smallest permutation representation of C4×S3×C3⋊S3
On 72 points
Generators in S72
(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 29 21)(2 30 22)(3 31 23)(4 32 24)(5 58 16)(6 59 13)(7 60 14)(8 57 15)(9 69 50)(10 70 51)(11 71 52)(12 72 49)(17 54 46)(18 55 47)(19 56 48)(20 53 45)(25 63 66)(26 64 67)(27 61 68)(28 62 65)(33 44 40)(34 41 37)(35 42 38)(36 43 39)
(1 18)(2 19)(3 20)(4 17)(5 69)(6 70)(7 71)(8 72)(9 58)(10 59)(11 60)(12 57)(13 51)(14 52)(15 49)(16 50)(21 55)(22 56)(23 53)(24 54)(25 42)(26 43)(27 44)(28 41)(29 47)(30 48)(31 45)(32 46)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
(1 16 33)(2 13 34)(3 14 35)(4 15 36)(5 44 29)(6 41 30)(7 42 31)(8 43 32)(9 68 55)(10 65 56)(11 66 53)(12 67 54)(17 49 64)(18 50 61)(19 51 62)(20 52 63)(21 58 40)(22 59 37)(23 60 38)(24 57 39)(25 45 71)(26 46 72)(27 47 69)(28 48 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 69 50)(10 70 51)(11 71 52)(12 72 49)(17 54 46)(18 55 47)(19 56 48)(20 53 45)(25 63 66)(26 64 67)(27 61 68)(28 62 65)(33 40 44)(34 37 41)(35 38 42)(36 39 43)
(1 20)(2 17)(3 18)(4 19)(5 66)(6 67)(7 68)(8 65)(9 42)(10 43)(11 44)(12 41)(13 64)(14 61)(15 62)(16 63)(21 45)(22 46)(23 47)(24 48)(25 58)(26 59)(27 60)(28 57)(29 53)(30 54)(31 55)(32 56)(33 52)(34 49)(35 50)(36 51)(37 72)(38 69)(39 70)(40 71)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S 12A ··· 12J 12K ··· 12R 12S ··· 12Z 12AA 12AB order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 12 12 size 1 1 3 3 9 9 27 27 2 ··· 2 4 4 4 4 1 1 3 3 9 9 27 27 2 ··· 2 4 4 4 4 6 ··· 6 18 18 2 ··· 2 4 ··· 4 6 ··· 6 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 D6 D6 C4×S3 C4×S3 S32 C2×S32 C4×S32 kernel C4×S3×C3⋊S3 S3×C3⋊Dic3 Dic3×C3⋊S3 C33⋊8(C2×C4) S3×C3×C12 C12×C3⋊S3 C4×C33⋊C2 C2×S3×C3⋊S3 S3×C3⋊S3 S3×C12 C4×C3⋊S3 C3×Dic3 C3⋊Dic3 C3×C12 S3×C6 C2×C3⋊S3 C3×S3 C3⋊S3 C12 C6 C3 # reps 1 1 1 1 1 1 1 1 8 4 1 4 1 5 4 1 16 4 4 4 8

Matrix representation of C4×S3×C3⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 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 12 0
,
 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 0 1 0 0 0 0 1 0
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C4×S3×C3⋊S3 in GAP, Magma, Sage, TeX

C_4\times S_3\times C_3\rtimes S_3
% in TeX

G:=Group("C4xS3xC3:S3");
// GroupNames label

G:=SmallGroup(432,670);
// by ID

G=gap.SmallGroup(432,670);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^3=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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