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## G = D9×C3⋊D4order 432 = 24·33

### Direct product of D9 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D9×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C2×S3×D9 — D9×C3⋊D4
 Lower central C3×C9 — C3×C18 — D9×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for D9×C3⋊D4
G = < a,b,c,d,e | a9=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1244 in 178 conjugacy classes, 45 normal (41 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C3×D9, C3×D9, S3×C9, C9⋊S3, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, C22×D9, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C2×S32, S3×C2×C6, C9×Dic3, C9⋊Dic3, S3×D9, C6×D9, C6×D9, S3×C18, C2×C9⋊S3, C6×C18, D4×D9, S3×C3⋊D4, Dic3×D9, C3⋊D36, D6⋊D9, C9×C3⋊D4, C6.D18, C2×S3×D9, C2×C6×D9, D9×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, S32, S3×D4, C2×C3⋊D4, C22×D9, C2×S32, S3×D9, D4×D9, S3×C3⋊D4, C2×S3×D9, D9×C3⋊D4

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12 18A 18B 18C 18D ··· 18O 18P 18Q 18R 36A 36B 36C order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 18 18 18 18 ··· 18 18 18 18 36 36 36 size 1 1 2 6 9 9 18 54 2 2 4 6 54 2 2 2 2 4 4 4 4 12 18 18 18 18 2 2 2 4 4 4 12 2 2 2 4 ··· 4 12 12 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D6 D9 C3⋊D4 D18 D18 D18 S32 S3×D4 C2×S32 S3×D9 D4×D9 S3×C3⋊D4 C2×S3×D9 D9×C3⋊D4 kernel D9×C3⋊D4 Dic3×D9 C3⋊D36 D6⋊D9 C9×C3⋊D4 C6.D18 C2×S3×D9 C2×C6×D9 C22×D9 C3×C3⋊D4 C3×D9 D18 C2×C18 C3×Dic3 S3×C6 C62 C3⋊D4 D9 Dic3 D6 C2×C6 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 3 4 3 3 3 1 1 1 3 3 2 3 6

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

 1 0 0 0 0 1 0 0 0 0 33 0 0 0 31 9
,
 36 0 0 0 0 36 0 0 0 0 28 1 0 0 31 9
,
 26 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 11 0 0 10 0 0 0 0 0 1 0 0 0 0 1
,
 0 11 0 0 27 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,33,31,0,0,0,9],[36,0,0,0,0,36,0,0,0,0,28,31,0,0,1,9],[26,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,11,0,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,11,0,0,0,0,0,1,0,0,0,0,1] >;

D9×C3⋊D4 in GAP, Magma, Sage, TeX

D_9\times C_3\rtimes D_4
% in TeX

G:=Group("D9xC3:D4");
// GroupNames label

G:=SmallGroup(432,314);
// by ID

G=gap.SmallGroup(432,314);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,3091,662,4037,7069]);
// Polycyclic

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

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