direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×SD32, D8.C6, C48⋊4C2, C16⋊2C6, Q16⋊1C6, C6.16D8, C12.37D4, C24.20C22, C8.3(C2×C6), C4.2(C3×D4), C2.4(C3×D8), (C3×Q16)⋊5C2, (C3×D8).2C2, SmallGroup(96,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×SD32
G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C3×SD32
►On 48 points
Generators in S48
(1 39 21)(2 40 22)(3 41 23)(4 42 24)(5 43 25)(6 44 26)(7 45 27)(8 46 28)(9 47 29)(10 48 30)(11 33 31)(12 34 32)(13 35 17)(14 36 18)(15 37 19)(16 38 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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)
G:=sub<Sym(48)| (1,39,21)(2,40,22)(3,41,23)(4,42,24)(5,43,25)(6,44,26)(7,45,27)(8,46,28)(9,47,29)(10,48,30)(11,33,31)(12,34,32)(13,35,17)(14,36,18)(15,37,19)(16,38,20), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)>;
G:=Group( (1,39,21)(2,40,22)(3,41,23)(4,42,24)(5,43,25)(6,44,26)(7,45,27)(8,46,28)(9,47,29)(10,48,30)(11,33,31)(12,34,32)(13,35,17)(14,36,18)(15,37,19)(16,38,20), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44) );
G=PermutationGroup([[(1,39,21),(2,40,22),(3,41,23),(4,42,24),(5,43,25),(6,44,26),(7,45,27),(8,46,28),(9,47,29),(10,48,30),(11,33,31),(12,34,32),(13,35,17),(14,36,18),(15,37,19),(16,38,20)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,45),(34,36),(35,43),(37,41),(38,48),(40,46),(42,44)]])
C3×SD32 is a maximal subgroup of
D48⋊C2 SD32⋊S3 D6.2D8
33 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 1 | 1 | 2 | 8 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D8 | C3×D4 | SD32 | C3×D8 | C3×SD32 |
| kernel | C3×SD32 | C48 | C3×D8 | C3×Q16 | SD32 | C16 | D8 | Q16 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×SD32 ►in GL2(𝔽7) generated by
| 2 | 0 |
| 0 | 2 |
| 0 | 1 |
| 1 | 6 |
| 1 | 6 |
| 0 | 6 |
G:=sub<GL(2,GF(7))| [2,0,0,2],[0,1,1,6],[1,0,6,6] >;
C3×SD32 in GAP, Magma, Sage, TeX
C_3\times {\rm SD}_{32} % in TeX
G:=Group("C3xSD32"); // GroupNames label
G:=SmallGroup(96,62);
// by ID
G=gap.SmallGroup(96,62);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,288,169,867,441,165,2164,1090,88]);
// 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^7>;
// generators/relations
Export