direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C16, C48⋊5C2, Dic3○C16, D6.2C8, C8.19D6, Dic3.2C8, C24.23C22, C16○(C3⋊C8), C3⋊C16⋊6C2, C3⋊1(C2×C16), C3⋊C8.3C4, C16○(C3⋊C16), C2.1(S3×C8), C6.1(C2×C8), (C4×S3).4C4, (S3×C8).3C2, C4.16(C4×S3), C12.21(C2×C4), SmallGroup(96,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C16 |
Generators and relations for S3×C16
G = < a,b,c | a16=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C16
►On 48 points
(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 30 43)(2 31 44)(3 32 45)(4 17 46)(5 18 47)(6 19 48)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(14 27 40)(15 28 41)(16 29 42)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 33)(29 34)(30 35)(31 36)(32 37)
G:=sub<Sym(48)| (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), (1,30,43)(2,31,44)(3,32,45)(4,17,46)(5,18,47)(6,19,48)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,28,41)(16,29,42), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,33)(29,34)(30,35)(31,36)(32,37)>;
G:=Group( (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), (1,30,43)(2,31,44)(3,32,45)(4,17,46)(5,18,47)(6,19,48)(7,20,33)(8,21,34)(9,22,35)(10,23,36)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,28,41)(16,29,42), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,33)(29,34)(30,35)(31,36)(32,37) );
G=PermutationGroup([[(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)], [(1,30,43),(2,31,44),(3,32,45),(4,17,46),(5,18,47),(6,19,48),(7,20,33),(8,21,34),(9,22,35),(10,23,36),(11,24,37),(12,25,38),(13,26,39),(14,27,40),(15,28,41),(16,29,42)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,33),(29,34),(30,35),(31,36),(32,37)]])
S3×C16 is a maximal subgroup of
C96⋊C2 D12.4C8 C16.12D6 D16⋊3S3 D6.2D8 D48⋊5C2 C24.60D6 D15⋊2C16 D15⋊C16
S3×C16 is a maximal quotient of
C96⋊C2 Dic3⋊C16 D6⋊C16 C24.60D6 D15⋊2C16 D15⋊C16
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 16A | ··· | 16H | 16I | ··· | 16P | 24A | 24B | 24C | 24D | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | C4×S3 | S3×C8 | S3×C16 |
| kernel | S3×C16 | C3⋊C16 | C48 | S3×C8 | C3⋊C8 | C4×S3 | Dic3 | D6 | S3 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of S3×C16 ►in GL2(𝔽17) generated by
| 7 | 0 |
| 0 | 7 |
| 0 | 9 |
| 15 | 16 |
| 1 | 9 |
| 0 | 16 |
G:=sub<GL(2,GF(17))| [7,0,0,7],[0,15,9,16],[1,0,9,16] >;
S3×C16 in GAP, Magma, Sage, TeX
S_3\times C_{16} % in TeX
G:=Group("S3xC16"); // GroupNames label
G:=SmallGroup(96,4);
// by ID
G=gap.SmallGroup(96,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,31,50,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^16=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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