direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C9⋊C8, C9⋊3C24, C36.6C6, C12.8D9, C18.3C12, C6.4Dic9, (C3×C9)⋊2C8, C4.2(C3×D9), C2.(C3×Dic9), C12.8(C3×S3), (C3×C36).4C2, (C3×C18).2C4, (C3×C12).15S3, C32.2(C3⋊C8), (C3×C6).6Dic3, C6.1(C3×Dic3), C3.1(C3×C3⋊C8), SmallGroup(216,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C3×C9⋊C8 |
Generators and relations for C3×C9⋊C8
G = < a,b,c | a3=b9=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C9⋊C8
►On 72 points
Generators in S72
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(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 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 68 31 50 13 59 22 41)(2 67 32 49 14 58 23 40)(3 66 33 48 15 57 24 39)(4 65 34 47 16 56 25 38)(5 64 35 46 17 55 26 37)(6 72 36 54 18 63 27 45)(7 71 28 53 10 62 19 44)(8 70 29 52 11 61 20 43)(9 69 30 51 12 60 21 42)
G:=sub<Sym(72)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(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), (1,2,3,4,5,6,7,8,9)(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), (1,68,31,50,13,59,22,41)(2,67,32,49,14,58,23,40)(3,66,33,48,15,57,24,39)(4,65,34,47,16,56,25,38)(5,64,35,46,17,55,26,37)(6,72,36,54,18,63,27,45)(7,71,28,53,10,62,19,44)(8,70,29,52,11,61,20,43)(9,69,30,51,12,60,21,42)>;
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(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), (1,2,3,4,5,6,7,8,9)(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), (1,68,31,50,13,59,22,41)(2,67,32,49,14,58,23,40)(3,66,33,48,15,57,24,39)(4,65,34,47,16,56,25,38)(5,64,35,46,17,55,26,37)(6,72,36,54,18,63,27,45)(7,71,28,53,10,62,19,44)(8,70,29,52,11,61,20,43)(9,69,30,51,12,60,21,42) );
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(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)], [(1,2,3,4,5,6,7,8,9),(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)], [(1,68,31,50,13,59,22,41),(2,67,32,49,14,58,23,40),(3,66,33,48,15,57,24,39),(4,65,34,47,16,56,25,38),(5,64,35,46,17,55,26,37),(6,72,36,54,18,63,27,45),(7,71,28,53,10,62,19,44),(8,70,29,52,11,61,20,43),(9,69,30,51,12,60,21,42)]])
C3×C9⋊C8 is a maximal subgroup of
C36.38D6 C36.40D6 D6.Dic9 C9⋊D24 C36.D6 C18.D12 C9⋊Dic12 D9×C24
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | D9 | C3×S3 | C3⋊C8 | Dic9 | C3×Dic3 | C3×D9 | C9⋊C8 | C3×C3⋊C8 | C3×Dic9 | C3×C9⋊C8 |
| kernel | C3×C9⋊C8 | C3×C36 | C9⋊C8 | C3×C18 | C36 | C3×C9 | C18 | C9 | C3×C12 | C3×C6 | C12 | C12 | C32 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 3 | 2 | 2 | 3 | 2 | 6 | 6 | 4 | 6 | 12 |
Matrix representation of C3×C9⋊C8 ►in GL2(𝔽37) generated by
| 26 | 0 |
| 0 | 26 |
| 34 | 0 |
| 0 | 12 |
| 0 | 31 |
| 1 | 0 |
G:=sub<GL(2,GF(37))| [26,0,0,26],[34,0,0,12],[0,1,31,0] >;
C3×C9⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_9\rtimes C_8
% in TeX
G:=Group("C3xC9:C8"); // GroupNames label
G:=SmallGroup(216,12);
// by ID
G=gap.SmallGroup(216,12);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,3604,208,5189]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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