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## G = D4×C9⋊S3order 432 = 24·33

### Direct product of D4 and C9⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C9⋊S3
G = < a,b,c,d,e | a4=b2=c9=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2000 in 270 conjugacy classes, 73 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, D4, C23, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×D4, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C3×D4, C22×S3, C3×C9, Dic9, C36, D18, C2×C18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×D4, C9⋊S3, C9⋊S3, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C22×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C2×C9⋊S3, C2×C9⋊S3, C6×C18, D4×D9, D4×C3⋊S3, C4×C9⋊S3, C36⋊S3, C6.D18, D4×C3×C9, C22×C9⋊S3, D4×C9⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, S3×D4, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, D4×D9, D4×C3⋊S3, C22×C9⋊S3, D4×C9⋊S3

Smallest permutation representation of D4×C9⋊S3
On 108 points
Generators in S108
(1 103 49 76)(2 104 50 77)(3 105 51 78)(4 106 52 79)(5 107 53 80)(6 108 54 81)(7 100 46 73)(8 101 47 74)(9 102 48 75)(10 59 86 32)(11 60 87 33)(12 61 88 34)(13 62 89 35)(14 63 90 36)(15 55 82 28)(16 56 83 29)(17 57 84 30)(18 58 85 31)(19 71 98 44)(20 72 99 45)(21 64 91 37)(22 65 92 38)(23 66 93 39)(24 67 94 40)(25 68 95 41)(26 69 96 42)(27 70 97 43)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 59)(11 60)(12 61)(13 62)(14 63)(15 55)(16 56)(17 57)(18 58)(19 71)(20 72)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 32 37)(2 33 38)(3 34 39)(4 35 40)(5 36 41)(6 28 42)(7 29 43)(8 30 44)(9 31 45)(10 21 103)(11 22 104)(12 23 105)(13 24 106)(14 25 107)(15 26 108)(16 27 100)(17 19 101)(18 20 102)(46 56 70)(47 57 71)(48 58 72)(49 59 64)(50 60 65)(51 61 66)(52 62 67)(53 63 68)(54 55 69)(73 83 97)(74 84 98)(75 85 99)(76 86 91)(77 87 92)(78 88 93)(79 89 94)(80 90 95)(81 82 96)
(2 9)(3 8)(4 7)(5 6)(10 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(28 41)(29 40)(30 39)(31 38)(32 37)(33 45)(34 44)(35 43)(36 42)(46 52)(47 51)(48 50)(53 54)(55 68)(56 67)(57 66)(58 65)(59 64)(60 72)(61 71)(62 70)(63 69)(73 79)(74 78)(75 77)(80 81)(82 95)(83 94)(84 93)(85 92)(86 91)(87 99)(88 98)(89 97)(90 96)(100 106)(101 105)(102 104)(107 108)

G:=sub<Sym(108)| (1,103,49,76)(2,104,50,77)(3,105,51,78)(4,106,52,79)(5,107,53,80)(6,108,54,81)(7,100,46,73)(8,101,47,74)(9,102,48,75)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(10,59)(11,60)(12,61)(13,62)(14,63)(15,55)(16,56)(17,57)(18,58)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,32,37)(2,33,38)(3,34,39)(4,35,40)(5,36,41)(6,28,42)(7,29,43)(8,30,44)(9,31,45)(10,21,103)(11,22,104)(12,23,105)(13,24,106)(14,25,107)(15,26,108)(16,27,100)(17,19,101)(18,20,102)(46,56,70)(47,57,71)(48,58,72)(49,59,64)(50,60,65)(51,61,66)(52,62,67)(53,63,68)(54,55,69)(73,83,97)(74,84,98)(75,85,99)(76,86,91)(77,87,92)(78,88,93)(79,89,94)(80,90,95)(81,82,96), (2,9)(3,8)(4,7)(5,6)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(28,41)(29,40)(30,39)(31,38)(32,37)(33,45)(34,44)(35,43)(36,42)(46,52)(47,51)(48,50)(53,54)(55,68)(56,67)(57,66)(58,65)(59,64)(60,72)(61,71)(62,70)(63,69)(73,79)(74,78)(75,77)(80,81)(82,95)(83,94)(84,93)(85,92)(86,91)(87,99)(88,98)(89,97)(90,96)(100,106)(101,105)(102,104)(107,108)>;

G:=Group( (1,103,49,76)(2,104,50,77)(3,105,51,78)(4,106,52,79)(5,107,53,80)(6,108,54,81)(7,100,46,73)(8,101,47,74)(9,102,48,75)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(10,59)(11,60)(12,61)(13,62)(14,63)(15,55)(16,56)(17,57)(18,58)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,32,37)(2,33,38)(3,34,39)(4,35,40)(5,36,41)(6,28,42)(7,29,43)(8,30,44)(9,31,45)(10,21,103)(11,22,104)(12,23,105)(13,24,106)(14,25,107)(15,26,108)(16,27,100)(17,19,101)(18,20,102)(46,56,70)(47,57,71)(48,58,72)(49,59,64)(50,60,65)(51,61,66)(52,62,67)(53,63,68)(54,55,69)(73,83,97)(74,84,98)(75,85,99)(76,86,91)(77,87,92)(78,88,93)(79,89,94)(80,90,95)(81,82,96), (2,9)(3,8)(4,7)(5,6)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(28,41)(29,40)(30,39)(31,38)(32,37)(33,45)(34,44)(35,43)(36,42)(46,52)(47,51)(48,50)(53,54)(55,68)(56,67)(57,66)(58,65)(59,64)(60,72)(61,71)(62,70)(63,69)(73,79)(74,78)(75,77)(80,81)(82,95)(83,94)(84,93)(85,92)(86,91)(87,99)(88,98)(89,97)(90,96)(100,106)(101,105)(102,104)(107,108) );

G=PermutationGroup([[(1,103,49,76),(2,104,50,77),(3,105,51,78),(4,106,52,79),(5,107,53,80),(6,108,54,81),(7,100,46,73),(8,101,47,74),(9,102,48,75),(10,59,86,32),(11,60,87,33),(12,61,88,34),(13,62,89,35),(14,63,90,36),(15,55,82,28),(16,56,83,29),(17,57,84,30),(18,58,85,31),(19,71,98,44),(20,72,99,45),(21,64,91,37),(22,65,92,38),(23,66,93,39),(24,67,94,40),(25,68,95,41),(26,69,96,42),(27,70,97,43)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,73),(8,74),(9,75),(10,59),(11,60),(12,61),(13,62),(14,63),(15,55),(16,56),(17,57),(18,58),(19,71),(20,72),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,32,37),(2,33,38),(3,34,39),(4,35,40),(5,36,41),(6,28,42),(7,29,43),(8,30,44),(9,31,45),(10,21,103),(11,22,104),(12,23,105),(13,24,106),(14,25,107),(15,26,108),(16,27,100),(17,19,101),(18,20,102),(46,56,70),(47,57,71),(48,58,72),(49,59,64),(50,60,65),(51,61,66),(52,62,67),(53,63,68),(54,55,69),(73,83,97),(74,84,98),(75,85,99),(76,86,91),(77,87,92),(78,88,93),(79,89,94),(80,90,95),(81,82,96)], [(2,9),(3,8),(4,7),(5,6),(10,21),(11,20),(12,19),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(28,41),(29,40),(30,39),(31,38),(32,37),(33,45),(34,44),(35,43),(36,42),(46,52),(47,51),(48,50),(53,54),(55,68),(56,67),(57,66),(58,65),(59,64),(60,72),(61,71),(62,70),(63,69),(73,79),(74,78),(75,77),(80,81),(82,95),(83,94),(84,93),(85,92),(86,91),(87,99),(88,98),(89,97),(90,96),(100,106),(101,105),(102,104),(107,108)]])

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E ··· 6L 9A ··· 9I 12A 12B 12C 12D 18A ··· 18I 18J ··· 18AA 36A ··· 36I order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 6 6 6 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 27 27 54 54 2 2 2 2 2 54 2 2 2 2 4 ··· 4 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D9 D18 D18 S3×D4 S3×D4 D4×D9 kernel D4×C9⋊S3 C4×C9⋊S3 C36⋊S3 C6.D18 D4×C3×C9 C22×C9⋊S3 D4×C9 D4×C32 C9⋊S3 C36 C2×C18 C3×C12 C62 C3×D4 C12 C2×C6 C9 C32 C3 # reps 1 1 1 2 1 2 3 1 2 3 6 1 2 9 9 18 3 1 9

Matrix representation of D4×C9⋊S3 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 35 1
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 0 1
,
 17 11 0 0 0 0 26 6 0 0 0 0 0 0 1 3 0 0 0 0 36 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 36 36 0 0 0 0 0 0 35 34 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 20 26 0 0 0 0 6 17 0 0 0 0 0 0 36 34 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,35,0,0,0,0,1,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,1,1],[17,26,0,0,0,0,11,6,0,0,0,0,0,0,1,36,0,0,0,0,3,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,36,0,0,0,0,0,0,35,1,0,0,0,0,34,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,6,0,0,0,0,26,17,0,0,0,0,0,0,36,0,0,0,0,0,34,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D4×C9⋊S3 in GAP, Magma, Sage, TeX

D_4\times C_9\rtimes S_3
% in TeX

G:=Group("D4xC9:S3");
// GroupNames label

G:=SmallGroup(432,388);
// by ID

G=gap.SmallGroup(432,388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,6164,662,4037,14118]);
// Polycyclic

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

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