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## G = C3×D4⋊D9order 432 = 24·33

### Direct product of C3 and D4⋊D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×D4⋊D9
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×D36 — C3×D4⋊D9
 Lower central C9 — C18 — C36 — C3×D4⋊D9
 Upper central C1 — C6 — C12 — 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=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 334 in 82 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, C3×C9, C36, C36, D18, C2×C18, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, C3×D9, C3×C18, C3×C18, C9⋊C8, D36, D4×C9, D4×C9, C3×C3⋊C8, C3×D12, D4×C32, C3×C36, C6×D9, C6×C18, D4⋊D9, C3×D4⋊S3, C3×C9⋊C8, C3×D36, D4×C3×C9, C3×D4⋊D9
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, D9, C3×S3, C3⋊D4, C3×D4, D18, S3×C6, D4⋊S3, C3×D8, C3×D9, C9⋊D4, C3×C3⋊D4, C6×D9, D4⋊D9, C3×D4⋊S3, C3×C9⋊D4, 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 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(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)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 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)| (1,4,7)(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), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(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), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(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( (1,4,7)(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), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(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), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(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([[(1,4,7),(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)], [(1,28,10,19),(2,29,11,20),(3,30,12,21),(4,31,13,22),(5,32,14,23),(6,33,15,24),(7,34,16,25),(8,35,17,26),(9,36,18,27),(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)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(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)]])

81 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F ··· 6M 6N 6O 8A 8B 9A ··· 9I 12A 12B 12C 12D 12E 18A ··· 18I 18J ··· 18AA 24A 24B 24C 24D 36A ··· 36I order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 ··· 6 6 6 8 8 9 ··· 9 12 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 size 1 1 4 36 1 1 2 2 2 2 1 1 2 2 2 4 ··· 4 36 36 18 18 2 ··· 2 2 2 4 4 4 2 ··· 2 4 ··· 4 18 18 18 18 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×D8 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 D4⋊S3 D4⋊D9 C3×D4⋊S3 C3×D4⋊D9 kernel C3×D4⋊D9 C3×C9⋊C8 C3×D36 D4×C3×C9 D4⋊D9 C9⋊C8 D36 D4×C9 D4×C32 C3×C18 C3×C12 C3×C9 C3×D4 C3×D4 C18 C3×C6 C12 C12 C9 D4 C6 C6 C4 C2 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 3 2 2 2 3 2 4 6 6 4 6 12 1 3 2 6

Matrix representation of C3×D4⋊D9 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
,
 55 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 0 69 0 0 18 0 0 0 0 0 16 57 0 0 57 57
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[55,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,18,0,0,69,0,0,0,0,0,16,57,0,0,57,57] >;

C3×D4⋊D9 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes D_9
% in TeX

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

G:=SmallGroup(432,149);
// by ID

G=gap.SmallGroup(432,149);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,10085,292,14118]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

׿
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