direct product, metacyclic, supersoluble, monomial
Aliases: C3xC8:S3, C24:5C6, C24:7S3, D6.C12, C12.65D6, Dic3.C12, C32:5M4(2), C3:C8:4C6, C8:3(C3xS3), (C3xC24):9C2, (C4xS3).2C6, (S3xC6).3C4, C2.3(S3xC12), C4.13(S3xC6), C6.22(C4xS3), C6.2(C2xC12), (S3xC12).5C2, C12.14(C2xC6), C3:1(C3xM4(2)), (C3xDic3).3C4, (C3xC12).43C22, (C3xC3:C8):11C2, (C3xC6).18(C2xC4), SmallGroup(144,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC8:S3
G = < a,b,c,d | a3=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C12, D6, C2xC6, M4(2), C3xS3, C4xS3, C2xC12, S3xC6, C8:S3, C3xM4(2), S3xC12, C3xC8:S3
Smallest permutation representation of C3xC8:S3
►On 48 points
Generators in S48
(1 30 40)(2 31 33)(3 32 34)(4 25 35)(5 26 36)(6 27 37)(7 28 38)(8 29 39)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(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)
(1 40 30)(2 33 31)(3 34 32)(4 35 25)(5 36 26)(6 37 27)(7 38 28)(8 39 29)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)
G:=sub<Sym(48)| (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (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), (1,40,30)(2,33,31)(3,34,32)(4,35,25)(5,36,26)(6,37,27)(7,38,28)(8,39,29)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44)>;
G:=Group( (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (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), (1,40,30)(2,33,31)(3,34,32)(4,35,25)(5,36,26)(6,37,27)(7,38,28)(8,39,29)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44) );
G=PermutationGroup([[(1,30,40),(2,31,33),(3,32,34),(4,25,35),(5,26,36),(6,27,37),(7,28,38),(8,29,39),(9,48,23),(10,41,24),(11,42,17),(12,43,18),(13,44,19),(14,45,20),(15,46,21),(16,47,22)], [(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)], [(1,40,30),(2,33,31),(3,34,32),(4,35,25),(5,36,26),(6,37,27),(7,38,28),(8,39,29),(9,48,23),(10,41,24),(11,42,17),(12,43,18),(13,44,19),(14,45,20),(15,46,21),(16,47,22)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,41),(26,46),(27,43),(28,48),(29,45),(30,42),(31,47),(32,44)]])
C3xC8:S3 is a maximal subgroup of
C24:D6 C24:1D6 D24:S3 C24.3D6 Dic12:S3 C24.64D6 C24.D6 C3xS3xM4(2) He3:5M4(2) C72:C6 He3:6M4(2)
C3xC8:S3 is a maximal quotient of
He3:5M4(2) C72:C6
54 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 24A | ··· | 24P | 24Q | 24R | 24S | 24T |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 2 | ··· | 2 | 6 | 6 | 6 | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | M4(2) | C3xS3 | C4xS3 | S3xC6 | C8:S3 | C3xM4(2) | S3xC12 | C3xC8:S3 |
| kernel | C3xC8:S3 | C3xC3:C8 | C3xC24 | S3xC12 | C8:S3 | C3xDic3 | S3xC6 | C3:C8 | C24 | C4xS3 | Dic3 | D6 | C24 | C12 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3xC8:S3 ►in GL4(F5) generated by
| 0 | 0 | 0 | 2 |
| 0 | 3 | 2 | 0 |
| 0 | 1 | 1 | 0 |
| 2 | 0 | 0 | 4 |
| 4 | 0 | 0 | 1 |
| 0 | 2 | 4 | 0 |
| 0 | 2 | 3 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 3 | 0 |
| 0 | 4 | 3 | 0 |
| 2 | 0 | 0 | 4 |
| 0 | 3 | 2 | 0 |
| 4 | 0 | 0 | 3 |
| 2 | 0 | 0 | 3 |
| 0 | 3 | 4 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,2,0,3,1,0,0,2,1,0,2,0,0,4],[4,0,0,1,0,2,2,0,0,4,3,0,1,0,0,1],[0,0,0,2,0,1,4,0,0,3,3,0,2,0,0,4],[0,4,2,0,3,0,0,3,2,0,0,4,0,3,3,0] >;
C3xC8:S3 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes S_3
% in TeX
G:=Group("C3xC8:S3"); // GroupNames label
G:=SmallGroup(144,70);
// by ID
G=gap.SmallGroup(144,70);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,313,79,69,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=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^5,d*c*d=c^-1>;
// generators/relations
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