direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×D16, C48⋊3C2, C16⋊1C6, D8⋊1C6, C6.15D8, C12.36D4, C24.19C22, (C3×D8)⋊5C2, C8.2(C2×C6), C4.1(C3×D4), C2.3(C3×D8), SmallGroup(96,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D16
G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D16
►On 48 points
Generators in S48
(1 17 38)(2 18 39)(3 19 40)(4 20 41)(5 21 42)(6 22 43)(7 23 44)(8 24 45)(9 25 46)(10 26 47)(11 27 48)(12 28 33)(13 29 34)(14 30 35)(15 31 36)(16 32 37)
(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 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)
G:=sub<Sym(48)| (1,17,38)(2,18,39)(3,19,40)(4,20,41)(5,21,42)(6,22,43)(7,23,44)(8,24,45)(9,25,46)(10,26,47)(11,27,48)(12,28,33)(13,29,34)(14,30,35)(15,31,36)(16,32,37), (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)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)>;
G:=Group( (1,17,38)(2,18,39)(3,19,40)(4,20,41)(5,21,42)(6,22,43)(7,23,44)(8,24,45)(9,25,46)(10,26,47)(11,27,48)(12,28,33)(13,29,34)(14,30,35)(15,31,36)(16,32,37), (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)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46) );
G=PermutationGroup([[(1,17,38),(2,18,39),(3,19,40),(4,20,41),(5,21,42),(6,22,43),(7,23,44),(8,24,45),(9,25,46),(10,26,47),(11,27,48),(12,28,33),(13,29,34),(14,30,35),(15,31,36),(16,32,37)], [(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),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(33,42),(34,41),(35,40),(36,39),(37,38),(43,48),(44,47),(45,46)]])
C3×D16 is a maximal subgroup of
C3⋊D32 D16.S3 D8⋊D6 D16⋊3S3
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 8 | 1 | 1 | 2 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | D16 | C3×D8 | C3×D16 |
| kernel | C3×D16 | C48 | C3×D8 | D16 | C16 | D8 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×D16 ►in GL2(𝔽31) generated by
| 25 | 0 |
| 0 | 25 |
| 14 | 25 |
| 26 | 0 |
| 0 | 25 |
| 5 | 0 |
G:=sub<GL(2,GF(31))| [25,0,0,25],[14,26,25,0],[0,5,25,0] >;
C3×D16 in GAP, Magma, Sage, TeX
C_3\times D_{16} % in TeX
G:=Group("C3xD16"); // GroupNames label
G:=SmallGroup(96,61);
// by ID
G=gap.SmallGroup(96,61);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,169,867,441,165,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export