direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3xC3:C16, C3:C48, C6.C24, C24.5C6, C32:3C16, C12.2C12, C24.10S3, C12.10Dic3, C6.4(C3:C8), C8.2(C3xS3), (C3xC6).3C8, (C3xC12).6C4, (C3xC24).4C2, C4.2(C3xDic3), C2.(C3xC3:C8), SmallGroup(144,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3xC3:C16 |
Generators and relations for C3xC3:C16
G = < a,b,c | a3=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C16, C3xS3, C3:C8, C24, C3xDic3, C3:C16, C48, C3xC3:C8, C3xC3:C16
Smallest permutation representation of C3xC3:C16
►On 48 points
(1 21 43)(2 22 44)(3 23 45)(4 24 46)(5 25 47)(6 26 48)(7 27 33)(8 28 34)(9 29 35)(10 30 36)(11 31 37)(12 32 38)(13 17 39)(14 18 40)(15 19 41)(16 20 42)
(1 21 43)(2 44 22)(3 23 45)(4 46 24)(5 25 47)(6 48 26)(7 27 33)(8 34 28)(9 29 35)(10 36 30)(11 31 37)(12 38 32)(13 17 39)(14 40 18)(15 19 41)(16 42 20)
(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)
G:=sub<Sym(48)| (1,21,43)(2,22,44)(3,23,45)(4,24,46)(5,25,47)(6,26,48)(7,27,33)(8,28,34)(9,29,35)(10,30,36)(11,31,37)(12,32,38)(13,17,39)(14,18,40)(15,19,41)(16,20,42), (1,21,43)(2,44,22)(3,23,45)(4,46,24)(5,25,47)(6,48,26)(7,27,33)(8,34,28)(9,29,35)(10,36,30)(11,31,37)(12,38,32)(13,17,39)(14,40,18)(15,19,41)(16,42,20), (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)>;
G:=Group( (1,21,43)(2,22,44)(3,23,45)(4,24,46)(5,25,47)(6,26,48)(7,27,33)(8,28,34)(9,29,35)(10,30,36)(11,31,37)(12,32,38)(13,17,39)(14,18,40)(15,19,41)(16,20,42), (1,21,43)(2,44,22)(3,23,45)(4,46,24)(5,25,47)(6,48,26)(7,27,33)(8,34,28)(9,29,35)(10,36,30)(11,31,37)(12,38,32)(13,17,39)(14,40,18)(15,19,41)(16,42,20), (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) );
G=PermutationGroup([[(1,21,43),(2,22,44),(3,23,45),(4,24,46),(5,25,47),(6,26,48),(7,27,33),(8,28,34),(9,29,35),(10,30,36),(11,31,37),(12,32,38),(13,17,39),(14,18,40),(15,19,41),(16,20,42)], [(1,21,43),(2,44,22),(3,23,45),(4,46,24),(5,25,47),(6,48,26),(7,27,33),(8,34,28),(9,29,35),(10,36,30),(11,31,37),(12,38,32),(13,17,39),(14,40,18),(15,19,41),(16,42,20)], [(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)]])
C3xC3:C16 is a maximal subgroup of
C24.60D6 C24.61D6 C24.62D6 C3:D48 C32:3SD32 C24.49D6 C32:3Q32 S3xC48 He3:3C16 C9:C48 He3:4C16
C3xC3:C16 is a maximal quotient of
He3:3C16 C9:C48
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24T | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | S3 | Dic3 | C3xS3 | C3:C8 | C3xDic3 | C3:C16 | C3xC3:C8 | C3xC3:C16 |
| kernel | C3xC3:C16 | C3xC24 | C3:C16 | C3xC12 | C24 | C3xC6 | C12 | C32 | C6 | C3 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3xC3:C16 ►in GL3(F97) generated by
| 35 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 35 |
| 85 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(97))| [35,0,0,0,61,0,0,0,61],[1,0,0,0,61,0,0,0,35],[85,0,0,0,0,1,0,1,0] >;
C3xC3:C16 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes C_{16} % in TeX
G:=Group("C3xC3:C16"); // GroupNames label
G:=SmallGroup(144,28);
// by ID
G=gap.SmallGroup(144,28);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,-3,36,50,69,3461]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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