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G = C3×D4×D9order 432 = 24·33

Direct product of C3, D4 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×D4×D9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C2×C6×D9 — C3×D4×D9
 Lower central C9 — C18 — C3×D4×D9
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4×D9
G = < a,b,c,d,e | a3=b4=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 750 in 178 conjugacy classes, 62 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, D9, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C3×C9, Dic9, C36, C36, D18, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×D4, C6×D4, C3×D9, C3×D9, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, D4×C9, C22×D9, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, C3×Dic9, C3×C36, C6×D9, C6×D9, C6×D9, C6×C18, D4×D9, C3×S3×D4, C12×D9, C3×D36, C3×C9⋊D4, D4×C3×C9, C2×C6×D9, C3×D4×D9
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, D9, C3×S3, C3×D4, C22×S3, C22×C6, D18, S3×C6, S3×D4, C6×D4, C3×D9, C22×D9, S3×C2×C6, C6×D9, D4×D9, C3×S3×D4, C2×C6×D9, C3×D4×D9

Smallest permutation representation of C3×D4×D9
On 72 points
Generators in S72
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 63)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 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 39)(2 38)(3 37)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 54)(18 53)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 63)(27 62)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 72)(36 71)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C ··· 6I 6J ··· 6O 6P 6Q 6R 6S 6T 6U 6V 6W 9A ··· 9I 12A 12B 12C 12D 12E 12F 12G 18A ··· 18I 18J ··· 18AA 36A ··· 36I order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 9 9 18 18 1 1 2 2 2 2 18 1 1 2 ··· 2 4 ··· 4 9 9 9 9 18 18 18 18 2 ··· 2 2 2 4 4 4 18 18 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D6 D6 D9 C3×S3 C3×D4 D18 S3×C6 D18 S3×C6 C3×D9 C6×D9 C6×D9 S3×D4 D4×D9 C3×S3×D4 C3×D4×D9 kernel C3×D4×D9 C12×D9 C3×D36 C3×C9⋊D4 D4×C3×C9 C2×C6×D9 D4×D9 C4×D9 D36 C9⋊D4 D4×C9 C22×D9 D4×C32 C3×D9 C3×C12 C62 C3×D4 C3×D4 D9 C12 C12 C2×C6 C2×C6 D4 C4 C22 C32 C3 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 2 1 2 3 2 4 3 2 6 4 6 6 12 1 3 2 6

Matrix representation of C3×D4×D9 in GL4(𝔽37) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 0 1 0 0 36 0
,
 36 0 0 0 0 36 0 0 0 0 0 1 0 0 1 0
,
 34 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 34 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,0,36,0,0,1,0],[36,0,0,0,0,36,0,0,0,0,0,1,0,0,1,0],[34,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,34,0,0,12,0,0,0,0,0,1,0,0,0,0,1] >;

C3×D4×D9 in GAP, Magma, Sage, TeX

C_3\times D_4\times D_9
% in TeX

G:=Group("C3xD4xD9");
// GroupNames label

G:=SmallGroup(432,356);
// by ID

G=gap.SmallGroup(432,356);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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