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## G = C3×D4⋊2D9order 432 = 24·33

### Direct product of C3 and D4⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×D4⋊2D9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C12×D9 — C3×D4⋊2D9
 Lower central C9 — C18 — C3×D4⋊2D9
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D42D9
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, ece=b2c, ede=d-1 >

Subgroups: 438 in 136 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, D42S3, C3×C4○D4, C3×D9, C3×C18, C3×C18, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, D4×C9, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, C3×Dic9, C3×Dic9, C3×C36, C6×D9, C6×C18, D42D9, C3×D42S3, C3×Dic18, C12×D9, C6×Dic9, C3×C9⋊D4, D4×C3×C9, C3×D42D9
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, D42S3, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9, D42D9, C3×D42S3, C2×C6×D9, C3×D42D9

Smallest permutation representation of C3×D42D9
On 72 points
Generators in S72
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 19 10 28)(2 20 11 29)(3 21 12 30)(4 22 13 31)(5 23 14 32)(6 24 15 33)(7 25 16 34)(8 26 17 35)(9 27 18 36)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)
(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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 45)(9 44)(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)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,19,10,28)(2,20,11,29)(3,21,12,30)(4,22,13,31)(5,23,14,32)(6,24,15,33)(7,25,16,34)(8,26,17,35)(9,27,18,36)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,45)(9,44)(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( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,19,10,28)(2,20,11,29)(3,21,12,30)(4,22,13,31)(5,23,14,32)(6,24,15,33)(7,25,16,34)(8,26,17,35)(9,27,18,36)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,45)(9,44)(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([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(1,19,10,28),(2,20,11,29),(3,21,12,30),(4,22,13,31),(5,23,14,32),(6,24,15,33),(7,25,16,34),(8,26,17,35),(9,27,18,36),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,45),(9,44),(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 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6I 6J ··· 6O 6P 6Q 9A ··· 9I 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M 18A ··· 18I 18J ··· 18AA 36A ··· 36I order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 9 ··· 9 12 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 18 1 1 2 2 2 2 9 9 18 18 1 1 2 ··· 2 4 ··· 4 18 18 2 ··· 2 2 2 4 4 4 9 9 9 9 18 18 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 D6 D6 C4○D4 D9 C3×S3 D18 S3×C6 D18 S3×C6 C3×C4○D4 C3×D9 C6×D9 C6×D9 D4⋊2S3 D4⋊2D9 C3×D4⋊2S3 C3×D4⋊2D9 kernel C3×D4⋊2D9 C3×Dic18 C12×D9 C6×Dic9 C3×C9⋊D4 D4×C3×C9 D4⋊2D9 Dic18 C4×D9 C2×Dic9 C9⋊D4 D4×C9 D4×C32 C3×C12 C62 C3×C9 C3×D4 C3×D4 C12 C12 C2×C6 C2×C6 C9 D4 C4 C22 C32 C3 C3 C1 # reps 1 1 1 2 2 1 2 2 2 4 4 2 1 1 2 2 3 2 3 2 6 4 4 6 6 12 1 3 2 6

Matrix representation of C3×D42D9 in GL4(𝔽37) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 36 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 36
,
 12 0 0 0 28 34 0 0 0 0 1 0 0 0 0 1
,
 29 36 0 0 26 8 0 0 0 0 0 31 0 0 6 0
G:=sub<GL(4,GF(37))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,36,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[12,28,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[29,26,0,0,36,8,0,0,0,0,0,6,0,0,31,0] >;

C3×D42D9 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2D_9
% in TeX

G:=Group("C3xD4:2D9");
// GroupNames label

G:=SmallGroup(432,357);
// by ID

G=gap.SmallGroup(432,357);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,10085,292,14118]);
// 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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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