direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3xC9:S3, C32:3D9, C33.6S3, C3:(C3xD9), C9:3(C3xS3), (C3xC9):15C6, (C3xC9):10S3, (C32xC9):4C2, C32.9(C3:S3), C32.16(C3xS3), C3.1(C3xC3:S3), SmallGroup(162,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC9 — C3xC9:S3 |
Generators and relations for C3xC9:S3
G = < a,b,c,d | a3=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 202 in 55 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3xS3, C3:S3, C3xC9, C3xC9, C3xC9, C33, C3xD9, C9:S3, C3xC3:S3, C32xC9, C3xC9:S3
Quotients: C1, C2, C3, S3, C6, D9, C3xS3, C3:S3, C3xD9, C9:S3, C3xC3:S3, C3xC9:S3
►On 54 points
(1 41 35)(2 42 36)(3 43 28)(4 44 29)(5 45 30)(6 37 31)(7 38 32)(8 39 33)(9 40 34)(10 53 22)(11 54 23)(12 46 24)(13 47 25)(14 48 26)(15 49 27)(16 50 19)(17 51 20)(18 52 21)
(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)
(1 38 29)(2 39 30)(3 40 31)(4 41 32)(5 42 33)(6 43 34)(7 44 35)(8 45 36)(9 37 28)(10 19 47)(11 20 48)(12 21 49)(13 22 50)(14 23 51)(15 24 52)(16 25 53)(17 26 54)(18 27 46)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(9 18)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(37 46)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)
G:=sub<Sym(54)| (1,41,35)(2,42,36)(3,43,28)(4,44,29)(5,45,30)(6,37,31)(7,38,32)(8,39,33)(9,40,34)(10,53,22)(11,54,23)(12,46,24)(13,47,25)(14,48,26)(15,49,27)(16,50,19)(17,51,20)(18,52,21), (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), (1,38,29)(2,39,30)(3,40,31)(4,41,32)(5,42,33)(6,43,34)(7,44,35)(8,45,36)(9,37,28)(10,19,47)(11,20,48)(12,21,49)(13,22,50)(14,23,51)(15,24,52)(16,25,53)(17,26,54)(18,27,46), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(9,18)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)>;
G:=Group( (1,41,35)(2,42,36)(3,43,28)(4,44,29)(5,45,30)(6,37,31)(7,38,32)(8,39,33)(9,40,34)(10,53,22)(11,54,23)(12,46,24)(13,47,25)(14,48,26)(15,49,27)(16,50,19)(17,51,20)(18,52,21), (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), (1,38,29)(2,39,30)(3,40,31)(4,41,32)(5,42,33)(6,43,34)(7,44,35)(8,45,36)(9,37,28)(10,19,47)(11,20,48)(12,21,49)(13,22,50)(14,23,51)(15,24,52)(16,25,53)(17,26,54)(18,27,46), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(9,18)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47) );
G=PermutationGroup([[(1,41,35),(2,42,36),(3,43,28),(4,44,29),(5,45,30),(6,37,31),(7,38,32),(8,39,33),(9,40,34),(10,53,22),(11,54,23),(12,46,24),(13,47,25),(14,48,26),(15,49,27),(16,50,19),(17,51,20),(18,52,21)], [(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)], [(1,38,29),(2,39,30),(3,40,31),(4,41,32),(5,42,33),(6,43,34),(7,44,35),(8,45,36),(9,37,28),(10,19,47),(11,20,48),(12,21,49),(13,22,50),(14,23,51),(15,24,52),(16,25,53),(17,26,54),(18,27,46)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(9,18),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(37,46),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47)]])
C3xC9:S3 is a maximal subgroup of
C3xS3xD9 C32:5D18 C9:S3:C9 C32:D27 (C3xC9):C18 C9:S3:3C9 C3.2(C9:D9) C32:2D27 (C3xC9):5D9 (C3xC9):6D9 C33.D9 He3:2D9 3- 1+2:D9 C9:(S3xC9) C92:3S3 C92:3C6 He3:3D9 C92:9C6 C33.5D9 C33:6D9 He3:4D9 He3.(C3:S3) C3:(He3:S3) (C32xC9).S3 C3wrC3:S3 C9oHe3:3S3
C3xC9:S3 is a maximal quotient of
C33:D9 C92:4S3 C92:3C6 He3:3D9 C92:9C6 C33.5D9 He3.5D9
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 9A | ··· | 9AA |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 |
| size | 1 | 27 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3xS3 | D9 | C3xS3 | C3xD9 |
| kernel | C3xC9:S3 | C32xC9 | C9:S3 | C3xC9 | C3xC9 | C33 | C9 | C32 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 9 | 2 | 18 |
Matrix representation of C3xC9:S3 ►in GL4(F19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 5 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[5,0,0,0,0,4,0,0,0,0,11,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3xC9:S3 in GAP, Magma, Sage, TeX
C_3\times C_9\rtimes S_3
% in TeX
G:=Group("C3xC9:S3"); // GroupNames label
G:=SmallGroup(162,38);
// by ID
G=gap.SmallGroup(162,38);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,992,282,723,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations