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## G = C9×D4⋊S3order 432 = 24·33

### Direct product of C9 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×D4⋊S3
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×D12 — C9×D4⋊S3
 Lower central C3 — C6 — C12 — C9×D4⋊S3
 Upper central C1 — C18 — C36 — D4×C9

Generators and relations for C9×D4⋊S3
G = < a,b,c,d,e | a9=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 208 in 82 conjugacy classes, 33 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, C18, C18, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, C3×C9, C36, C36, C2×C18, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, S3×C9, C3×C18, C3×C18, C72, D4×C9, D4×C9, C3×C3⋊C8, C3×D12, D4×C32, C3×C36, S3×C18, C6×C18, C9×D8, C3×D4⋊S3, C9×C3⋊C8, C9×D12, D4×C3×C9, C9×D4⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, D8, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, D4⋊S3, C3×D8, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×D8, C3×D4⋊S3, C9×C3⋊D4, C9×D4⋊S3

Smallest permutation representation of C9×D4⋊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 53 31 38)(2 54 32 39)(3 46 33 40)(4 47 34 41)(5 48 35 42)(6 49 36 43)(7 50 28 44)(8 51 29 45)(9 52 30 37)(10 63 67 22)(11 55 68 23)(12 56 69 24)(13 57 70 25)(14 58 71 26)(15 59 72 27)(16 60 64 19)(17 61 65 20)(18 62 66 21)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(28 50)(29 51)(30 52)(31 53)(32 54)(33 46)(34 47)(35 48)(36 49)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 50)(20 51)(21 52)(22 53)(23 54)(24 46)(25 47)(26 48)(27 49)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 62)(38 63)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)

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

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

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

108 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 ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 18A ··· 18F 18G ··· 18L 18M ··· 18AD 18AE ··· 18AJ 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 ··· 6 6 6 8 8 9 ··· 9 9 ··· 9 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 4 12 1 1 2 2 2 2 1 1 2 2 2 4 ··· 4 12 12 6 6 1 ··· 1 2 ··· 2 2 2 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 12 ··· 12 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

108 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 2 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 S3×C9 D4×C9 C3×C3⋊D4 S3×C18 C9×D8 C9×C3⋊D4 D4⋊S3 C3×D4⋊S3 C9×D4⋊S3 kernel C9×D4⋊S3 C9×C3⋊C8 C9×D12 D4×C3×C9 C3×D4⋊S3 C3×C3⋊C8 C3×D12 D4×C32 D4⋊S3 C3⋊C8 D12 C3×D4 D4×C9 C3×C18 C36 C3×C9 C3×D4 C18 C3×C6 C12 C32 D4 C6 C6 C4 C3 C2 C9 C3 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 2 2 2 2 2 4 6 6 4 6 12 12 1 2 6

Matrix representation of C9×D4⋊S3 in GL4(𝔽73) generated by

 4 0 0 0 0 4 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 8 0 0 0 8 64 0 0 0 0 1 0 0 0 0 1
,
 1 7 0 0 0 72 0 0 0 0 16 57 0 0 57 57
G:=sub<GL(4,GF(73))| [4,0,0,0,0,4,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[8,8,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,7,72,0,0,0,0,16,57,0,0,57,57] >;

C9×D4⋊S3 in GAP, Magma, Sage, TeX

C_9\times D_4\rtimes S_3
% in TeX

G:=Group("C9xD4:S3");
// GroupNames label

G:=SmallGroup(432,150);
// by ID

G=gap.SmallGroup(432,150);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,142,2355,1186,192,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^4=c^2=d^3=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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