direct product, metacyclic, nilpotent (class 2), monomial
Aliases: D4×C3×C9, C36⋊7C6, C12⋊3C18, C6.8C62, C62.9C6, C4⋊(C3×C18), (C2×C6)⋊3C18, (C3×C36)⋊7C2, (C2×C18)⋊9C6, (C6×C18)⋊2C2, C2.1(C6×C18), C6.8(C2×C18), C12.3(C3×C6), C18.16(C2×C6), (C3×C12).17C6, C22⋊3(C3×C18), C32.4(C3×D4), C3.1(D4×C32), (C3×D4).1C32, (D4×C32).2C3, (C3×C18).23C22, (C2×C6).6(C3×C6), (C3×C6).34(C2×C6), (C3×C18)○(D4×C32), SmallGroup(216,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C3×C9
G = < a,b,c,d | a3=b9=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 100 in 80 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, D4, C9, C32, C12, C12, C2×C6, C18, C18, C3×C6, C3×C6, C3×D4, C3×D4, C3×C9, C36, C2×C18, C3×C12, C62, C3×C18, C3×C18, D4×C9, D4×C32, C3×C36, C6×C18, D4×C3×C9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2×C6, C18, C3×C6, C3×D4, C3×C9, C2×C18, C62, C3×C18, D4×C9, D4×C32, C6×C18, D4×C3×C9
►On 108 points
Generators in S108
(1 59 16)(2 60 17)(3 61 18)(4 62 10)(5 63 11)(6 55 12)(7 56 13)(8 57 14)(9 58 15)(19 72 103)(20 64 104)(21 65 105)(22 66 106)(23 67 107)(24 68 108)(25 69 100)(26 70 101)(27 71 102)(28 51 80)(29 52 81)(30 53 73)(31 54 74)(32 46 75)(33 47 76)(34 48 77)(35 49 78)(36 50 79)(37 97 87)(38 98 88)(39 99 89)(40 91 90)(41 92 82)(42 93 83)(43 94 84)(44 95 85)(45 96 86)
(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)
(1 101 36 97)(2 102 28 98)(3 103 29 99)(4 104 30 91)(5 105 31 92)(6 106 32 93)(7 107 33 94)(8 108 34 95)(9 100 35 96)(10 64 73 40)(11 65 74 41)(12 66 75 42)(13 67 76 43)(14 68 77 44)(15 69 78 45)(16 70 79 37)(17 71 80 38)(18 72 81 39)(19 52 89 61)(20 53 90 62)(21 54 82 63)(22 46 83 55)(23 47 84 56)(24 48 85 57)(25 49 86 58)(26 50 87 59)(27 51 88 60)
(19 89)(20 90)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(37 70)(38 71)(39 72)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(91 104)(92 105)(93 106)(94 107)(95 108)(96 100)(97 101)(98 102)(99 103)
G:=sub<Sym(108)| (1,59,16)(2,60,17)(3,61,18)(4,62,10)(5,63,11)(6,55,12)(7,56,13)(8,57,14)(9,58,15)(19,72,103)(20,64,104)(21,65,105)(22,66,106)(23,67,107)(24,68,108)(25,69,100)(26,70,101)(27,71,102)(28,51,80)(29,52,81)(30,53,73)(31,54,74)(32,46,75)(33,47,76)(34,48,77)(35,49,78)(36,50,79)(37,97,87)(38,98,88)(39,99,89)(40,91,90)(41,92,82)(42,93,83)(43,94,84)(44,95,85)(45,96,86), (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), (1,101,36,97)(2,102,28,98)(3,103,29,99)(4,104,30,91)(5,105,31,92)(6,106,32,93)(7,107,33,94)(8,108,34,95)(9,100,35,96)(10,64,73,40)(11,65,74,41)(12,66,75,42)(13,67,76,43)(14,68,77,44)(15,69,78,45)(16,70,79,37)(17,71,80,38)(18,72,81,39)(19,52,89,61)(20,53,90,62)(21,54,82,63)(22,46,83,55)(23,47,84,56)(24,48,85,57)(25,49,86,58)(26,50,87,59)(27,51,88,60), (19,89)(20,90)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(37,70)(38,71)(39,72)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(91,104)(92,105)(93,106)(94,107)(95,108)(96,100)(97,101)(98,102)(99,103)>;
G:=Group( (1,59,16)(2,60,17)(3,61,18)(4,62,10)(5,63,11)(6,55,12)(7,56,13)(8,57,14)(9,58,15)(19,72,103)(20,64,104)(21,65,105)(22,66,106)(23,67,107)(24,68,108)(25,69,100)(26,70,101)(27,71,102)(28,51,80)(29,52,81)(30,53,73)(31,54,74)(32,46,75)(33,47,76)(34,48,77)(35,49,78)(36,50,79)(37,97,87)(38,98,88)(39,99,89)(40,91,90)(41,92,82)(42,93,83)(43,94,84)(44,95,85)(45,96,86), (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), (1,101,36,97)(2,102,28,98)(3,103,29,99)(4,104,30,91)(5,105,31,92)(6,106,32,93)(7,107,33,94)(8,108,34,95)(9,100,35,96)(10,64,73,40)(11,65,74,41)(12,66,75,42)(13,67,76,43)(14,68,77,44)(15,69,78,45)(16,70,79,37)(17,71,80,38)(18,72,81,39)(19,52,89,61)(20,53,90,62)(21,54,82,63)(22,46,83,55)(23,47,84,56)(24,48,85,57)(25,49,86,58)(26,50,87,59)(27,51,88,60), (19,89)(20,90)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(37,70)(38,71)(39,72)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(91,104)(92,105)(93,106)(94,107)(95,108)(96,100)(97,101)(98,102)(99,103) );
G=PermutationGroup([[(1,59,16),(2,60,17),(3,61,18),(4,62,10),(5,63,11),(6,55,12),(7,56,13),(8,57,14),(9,58,15),(19,72,103),(20,64,104),(21,65,105),(22,66,106),(23,67,107),(24,68,108),(25,69,100),(26,70,101),(27,71,102),(28,51,80),(29,52,81),(30,53,73),(31,54,74),(32,46,75),(33,47,76),(34,48,77),(35,49,78),(36,50,79),(37,97,87),(38,98,88),(39,99,89),(40,91,90),(41,92,82),(42,93,83),(43,94,84),(44,95,85),(45,96,86)], [(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)], [(1,101,36,97),(2,102,28,98),(3,103,29,99),(4,104,30,91),(5,105,31,92),(6,106,32,93),(7,107,33,94),(8,108,34,95),(9,100,35,96),(10,64,73,40),(11,65,74,41),(12,66,75,42),(13,67,76,43),(14,68,77,44),(15,69,78,45),(16,70,79,37),(17,71,80,38),(18,72,81,39),(19,52,89,61),(20,53,90,62),(21,54,82,63),(22,46,83,55),(23,47,84,56),(24,48,85,57),(25,49,86,58),(26,50,87,59),(27,51,88,60)], [(19,89),(20,90),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(37,70),(38,71),(39,72),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(91,104),(92,105),(93,106),(94,107),(95,108),(96,100),(97,101),(98,102),(99,103)]])
D4×C3×C9 is a maximal subgroup of
C36.17D6 C36.18D6 C36.27D6
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4 | 6A | ··· | 6H | 6I | ··· | 6X | 9A | ··· | 9R | 12A | ··· | 12H | 18A | ··· | 18R | 18S | ··· | 18BB | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | D4 | C3×D4 | C3×D4 | D4×C9 |
| kernel | D4×C3×C9 | C3×C36 | C6×C18 | D4×C9 | D4×C32 | C36 | C2×C18 | C3×C12 | C62 | C3×D4 | C12 | C2×C6 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 12 | 2 | 4 | 18 | 18 | 36 | 1 | 6 | 2 | 18 |
Matrix representation of D4×C3×C9 ►in GL3(𝔽37) generated by
| 26 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 26 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 36 | 0 | 0 |
| 0 | 0 | 36 |
| 0 | 1 | 0 |
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 36 |
G:=sub<GL(3,GF(37))| [26,0,0,0,26,0,0,0,26],[1,0,0,0,12,0,0,0,12],[36,0,0,0,0,1,0,36,0],[36,0,0,0,1,0,0,0,36] >;
D4×C3×C9 in GAP, Magma, Sage, TeX
D_4\times C_3\times C_9
% in TeX
G:=Group("D4xC3xC9"); // GroupNames label
G:=SmallGroup(216,76);
// by ID
G=gap.SmallGroup(216,76);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,338]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations