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## G = C9×D4⋊2S3order 432 = 24·33

### Direct product of C9 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×D4⋊2S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C36 — C9×D4⋊2S3
 Lower central C3 — C6 — C9×D4⋊2S3
 Upper central C1 — C18 — D4×C9

Generators and relations for C9×D42S3
G = < a,b,c,d,e | a9=b4=c2=d3=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: 256 in 136 conjugacy classes, 69 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, 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, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, D42S3, C3×C4○D4, S3×C9, C3×C18, C3×C18, C2×C36, D4×C9, D4×C9, Q8×C9, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, C9×Dic3, C9×Dic3, C3×C36, S3×C18, C6×C18, C9×C4○D4, C3×D42S3, C9×Dic6, S3×C36, Dic3×C18, C9×C3⋊D4, D4×C3×C9, C9×D42S3
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2×C6, C4○D4, C18, C3×S3, C22×S3, C22×C6, C2×C18, S3×C6, D42S3, C3×C4○D4, S3×C9, C22×C18, S3×C2×C6, S3×C18, C9×C4○D4, C3×D42S3, S3×C2×C18, C9×D42S3

Smallest permutation representation of C9×D42S3
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 31 20 38)(2 32 21 39)(3 33 22 40)(4 34 23 41)(5 35 24 42)(6 36 25 43)(7 28 26 44)(8 29 27 45)(9 30 19 37)(10 49 67 56)(11 50 68 57)(12 51 69 58)(13 52 70 59)(14 53 71 60)(15 54 72 61)(16 46 64 62)(17 47 65 63)(18 48 66 55)
(28 44)(29 45)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(46 62)(47 63)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 37)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)

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

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

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

135 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 ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M 18A ··· 18F 18G ··· 18X 18Y ··· 18AJ 18AK ··· 18AP 36A ··· 36F 36G ··· 36R 36S ··· 36X 36Y ··· 36AJ 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 9 ··· 9 12 12 12 12 12 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 2 2 6 1 1 2 2 2 2 3 3 6 6 1 1 2 ··· 2 4 ··· 4 6 6 1 ··· 1 2 ··· 2 2 2 3 3 3 3 4 4 4 6 6 6 6 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C9 C18 C18 C18 C18 C18 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 S3×C9 S3×C18 S3×C18 C9×C4○D4 D4⋊2S3 C3×D4⋊2S3 C9×D4⋊2S3 kernel C9×D4⋊2S3 C9×Dic6 S3×C36 Dic3×C18 C9×C3⋊D4 D4×C3×C9 C3×D4⋊2S3 C3×Dic6 S3×C12 C6×Dic3 C3×C3⋊D4 D4×C32 D4⋊2S3 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 D4×C9 C36 C2×C18 C3×C9 C3×D4 C12 C2×C6 C32 D4 C4 C22 C3 C9 C3 C1 # reps 1 1 1 2 2 1 2 2 2 4 4 2 6 6 6 12 12 6 1 1 2 2 2 2 4 4 6 6 12 12 1 2 6

Matrix representation of C9×D42S3 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 21 2 0 0 1 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 16 36 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 6 26
,
 15 12 0 0 6 22 0 0 0 0 8 9 0 0 30 29
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[21,1,0,0,2,16,0,0,0,0,1,0,0,0,0,1],[1,16,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,6,0,0,0,26],[15,6,0,0,12,22,0,0,0,0,8,30,0,0,9,29] >;

C9×D42S3 in GAP, Magma, Sage, TeX

C_9\times D_4\rtimes_2S_3
% in TeX

G:=Group("C9xD4:2S3");
// GroupNames label

G:=SmallGroup(432,359);
// by ID

G=gap.SmallGroup(432,359);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,192,14118]);
// 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=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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