metabelian, supersoluble, monomial
Aliases: C9⋊S3⋊3C9, (C3×C9)⋊3C18, C32⋊C9.6S3, (C32×C9).5C6, C32.8(S3×C9), C33.51(C3×S3), C3.4(C32⋊C18), C32.20He3⋊1C2, C32.38(C32⋊C6), C3.4(He3.2S3), (C3×C9⋊S3).3C3, SmallGroup(486,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C9⋊S3⋊3C9 |
Generators and relations for C9⋊S3⋊3C9
G = < a,b,c,d | a9=b3=c2=d9=1, ab=ba, cac=a-1, dad-1=a7b, cbc=b-1, dbd-1=a6b, cd=dc >
Smallest permutation representation of C9⋊S3⋊3C9
►On 54 points
Generators in S54
(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)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 54)
(1 23 17 4 26 11 7 20 14)(2 27 18 5 21 12 8 24 15)(3 22 10 6 25 13 9 19 16)(28 46 43 34 52 40 31 49 37)(29 50 44 35 47 41 32 53 38)(30 54 45 36 51 42 33 48 39)
G:=sub<Sym(54)| (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), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,31)(2,30)(3,29)(4,28)(5,36)(6,35)(7,34)(8,33)(9,32)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,54), (1,23,17,4,26,11,7,20,14)(2,27,18,5,21,12,8,24,15)(3,22,10,6,25,13,9,19,16)(28,46,43,34,52,40,31,49,37)(29,50,44,35,47,41,32,53,38)(30,54,45,36,51,42,33,48,39)>;
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), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,31)(2,30)(3,29)(4,28)(5,36)(6,35)(7,34)(8,33)(9,32)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,54), (1,23,17,4,26,11,7,20,14)(2,27,18,5,21,12,8,24,15)(3,22,10,6,25,13,9,19,16)(28,46,43,34,52,40,31,49,37)(29,50,44,35,47,41,32,53,38)(30,54,45,36,51,42,33,48,39) );
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)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,31),(2,30),(3,29),(4,28),(5,36),(6,35),(7,34),(8,33),(9,32),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,45),(19,53),(20,52),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,54)], [(1,23,17,4,26,11,7,20,14),(2,27,18,5,21,12,8,24,15),(3,22,10,6,25,13,9,19,16),(28,46,43,34,52,40,31,49,37),(29,50,44,35,47,41,32,53,38),(30,54,45,36,51,42,33,48,39)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9O | 9P | ··· | 9U | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 27 | 27 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | C32⋊C6 | C32⋊C18 | He3.2S3 | C9⋊S3⋊3C9 |
| kernel | C9⋊S3⋊3C9 | C32.20He3 | C3×C9⋊S3 | C32×C9 | C9⋊S3 | C3×C9 | C32⋊C9 | C33 | C32 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of C9⋊S3⋊3C9 ►in GL6(𝔽19)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 7 | 0 | 0 |
G:=sub<GL(6,GF(19))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0] >;
C9⋊S3⋊3C9 in GAP, Magma, Sage, TeX
C_9\rtimes S_3\rtimes_3C_9
% in TeX
G:=Group("C9:S3:3C9"); // GroupNames label
G:=SmallGroup(486,22);
// by ID
G=gap.SmallGroup(486,22);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,873,453,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=c^2=d^9=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^7*b,c*b*c=b^-1,d*b*d^-1=a^6*b,c*d=d*c>;
// generators/relations
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