direct product, metabelian, supersoluble, monomial
Aliases: C3×C3⋊D16, D24⋊3C6, C32⋊6D16, C24.52D6, C3⋊C16⋊1C6, (C3×D8)⋊1C6, (C3×D8)⋊5S3, D8⋊1(C3×S3), C3⋊2(C3×D16), C8.4(S3×C6), C6.8(C3×D8), (C3×D24)⋊7C2, C24.2(C2×C6), (C3×C6).30D8, C12.3(C3×D4), (C3×C12).41D4, (C32×D8)⋊1C2, C6.30(D4⋊S3), C12.83(C3⋊D4), (C3×C24).13C22, (C3×C3⋊C16)⋊4C2, C2.4(C3×D4⋊S3), C4.1(C3×C3⋊D4), SmallGroup(288,260)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C3⋊D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 250 in 67 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, C16, D8, D8, C3×S3, C3×C6, C3×C6, C24, C24, D12, C3×D4, D16, C3×C12, S3×C6, C62, C3⋊C16, C48, D24, C3×D8, C3×D8, C3×C24, C3×D12, D4×C32, C3⋊D16, C3×D16, C3×C3⋊C16, C3×D24, C32×D8, C3×C3⋊D16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, D16, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C3⋊D16, C3×D16, C3×D4⋊S3, C3×C3⋊D16
►On 48 points
(1 40 23)(2 41 24)(3 42 25)(4 43 26)(5 44 27)(6 45 28)(7 46 29)(8 47 30)(9 48 31)(10 33 32)(11 34 17)(12 35 18)(13 36 19)(14 37 20)(15 38 21)(16 39 22)
(1 40 23)(2 24 41)(3 42 25)(4 26 43)(5 44 27)(6 28 45)(7 46 29)(8 30 47)(9 48 31)(10 32 33)(11 34 17)(12 18 35)(13 36 19)(14 20 37)(15 38 21)(16 22 39)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 32)(30 31)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 48)
G:=sub<Sym(48)| (1,40,23)(2,41,24)(3,42,25)(4,43,26)(5,44,27)(6,45,28)(7,46,29)(8,47,30)(9,48,31)(10,33,32)(11,34,17)(12,35,18)(13,36,19)(14,37,20)(15,38,21)(16,39,22), (1,40,23)(2,24,41)(3,42,25)(4,26,43)(5,44,27)(6,28,45)(7,46,29)(8,30,47)(9,48,31)(10,32,33)(11,34,17)(12,18,35)(13,36,19)(14,20,37)(15,38,21)(16,22,39), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,32)(30,31)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48)>;
G:=Group( (1,40,23)(2,41,24)(3,42,25)(4,43,26)(5,44,27)(6,45,28)(7,46,29)(8,47,30)(9,48,31)(10,33,32)(11,34,17)(12,35,18)(13,36,19)(14,37,20)(15,38,21)(16,39,22), (1,40,23)(2,24,41)(3,42,25)(4,26,43)(5,44,27)(6,28,45)(7,46,29)(8,30,47)(9,48,31)(10,32,33)(11,34,17)(12,18,35)(13,36,19)(14,20,37)(15,38,21)(16,22,39), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,32)(30,31)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48) );
G=PermutationGroup([[(1,40,23),(2,41,24),(3,42,25),(4,43,26),(5,44,27),(6,45,28),(7,46,29),(8,47,30),(9,48,31),(10,33,32),(11,34,17),(12,35,18),(13,36,19),(14,37,20),(15,38,21),(16,39,22)], [(1,40,23),(2,24,41),(3,42,25),(4,26,43),(5,44,27),(6,28,45),(7,46,29),(8,30,47),(9,48,31),(10,32,33),(11,34,17),(12,18,35),(13,36,19),(14,20,37),(15,38,21),(16,22,39)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(29,32),(30,31),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(47,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6M | 6N | 6O | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 24 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | C3⋊D4 | C3×D4 | D16 | S3×C6 | C3×D8 | C3×C3⋊D4 | C3×D16 | D4⋊S3 | C3⋊D16 | C3×D4⋊S3 | C3×C3⋊D16 |
| kernel | C3×C3⋊D16 | C3×C3⋊C16 | C3×D24 | C32×D8 | C3⋊D16 | C3⋊C16 | D24 | C3×D8 | C3×D8 | C3×C12 | C24 | C3×C6 | D8 | C12 | C12 | C32 | C8 | C6 | C4 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C3⋊D16 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 2 | 6 | 3 |
| 6 | 3 | 3 | 2 |
| 0 | 0 | 2 | 0 |
| 5 | 5 | 1 | 1 |
| 0 | 1 | 6 | 2 |
| 4 | 0 | 0 | 3 |
| 4 | 4 | 5 | 2 |
| 5 | 4 | 3 | 2 |
| 1 | 0 | 6 | 3 |
| 0 | 2 | 3 | 2 |
| 0 | 3 | 2 | 2 |
| 0 | 1 | 1 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,6,0,5,2,3,0,5,6,3,2,1,3,2,0,1],[0,4,4,5,1,0,4,4,6,0,5,3,2,3,2,2],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;
C3×C3⋊D16 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes D_{16} % in TeX
G:=Group("C3xC3:D16"); // GroupNames label
G:=SmallGroup(288,260);
// by ID
G=gap.SmallGroup(288,260);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,1011,514,192,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations