direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×SD32, C16⋊5D6, Q16⋊1D6, D8.2D6, C48⋊5C22, D6.13D8, Dic3.4D8, C24.16C23, D24.2C22, Dic12⋊5C22, (S3×D8).C2, C3⋊C8.13D4, C4.4(S3×D4), C3⋊2(C2×SD32), (S3×C16)⋊4C2, C3⋊C16⋊6C22, D8.S3⋊3C2, (S3×Q16)⋊3C2, C48⋊C2⋊5C2, C6.35(C2×D8), C2.19(S3×D8), (C3×SD32)⋊3C2, (C4×S3).20D4, C8.6D6⋊1C2, C12.10(C2×D4), C8.22(C22×S3), (C3×Q16)⋊4C22, (C3×D8).2C22, (S3×C8).11C22, SmallGroup(192,472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×SD32
G = < a,b,c,d | a3=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
Subgroups: 364 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×C16, SD32, SD32, C2×D8, C2×Q16, C3⋊C16, C48, S3×C8, D24, Dic12, D4⋊S3, C3⋊Q16, C3×D8, C3×Q16, S3×D4, S3×Q8, C2×SD32, S3×C16, C48⋊C2, D8.S3, C8.6D6, C3×SD32, S3×D8, S3×Q16, S3×SD32
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, SD32, C2×D8, S3×D4, C2×SD32, S3×D8, S3×SD32
►On 48 points
(1 44 25)(2 45 26)(3 46 27)(4 47 28)(5 48 29)(6 33 30)(7 34 31)(8 35 32)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 22)(15 42 23)(16 43 24)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 44)(18 45)(19 46)(20 47)(21 48)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)
(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)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 47)(34 38)(35 45)(37 43)(39 41)(40 48)(42 46)
G:=sub<Sym(48)| (1,44,25)(2,45,26)(3,46,27)(4,47,28)(5,48,29)(6,33,30)(7,34,31)(8,35,32)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,22)(15,42,23)(16,43,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43), (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)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46)>;
G:=Group( (1,44,25)(2,45,26)(3,46,27)(4,47,28)(5,48,29)(6,33,30)(7,34,31)(8,35,32)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,22)(15,42,23)(16,43,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43), (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)(18,24)(19,31)(20,22)(21,29)(23,27)(26,32)(28,30)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46) );
G=PermutationGroup([[(1,44,25),(2,45,26),(3,46,27),(4,47,28),(5,48,29),(6,33,30),(7,34,31),(8,35,32),(9,36,17),(10,37,18),(11,38,19),(12,39,20),(13,40,21),(14,41,22),(15,42,23),(16,43,24)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,44),(18,45),(19,46),(20,47),(21,48),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43)], [(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),(18,24),(19,31),(20,22),(21,29),(23,27),(26,32),(28,30),(33,47),(34,38),(35,45),(37,43),(39,41),(40,48),(42,46)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 48A | 48B | 48C | 48D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 48 | 48 | 48 | 48 |
| size | 1 | 1 | 3 | 3 | 8 | 24 | 2 | 2 | 6 | 8 | 24 | 2 | 16 | 2 | 2 | 6 | 6 | 4 | 16 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D8 | D8 | SD32 | S3×D4 | S3×D8 | S3×SD32 |
| kernel | S3×SD32 | S3×C16 | C48⋊C2 | D8.S3 | C8.6D6 | C3×SD32 | S3×D8 | S3×Q16 | SD32 | C3⋊C8 | C4×S3 | C16 | D8 | Q16 | Dic3 | D6 | S3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 4 |
Matrix representation of S3×SD32 ►in GL4(𝔽7) generated by
| 3 | 0 | 4 | 6 |
| 1 | 2 | 6 | 5 |
| 4 | 0 | 2 | 4 |
| 1 | 0 | 3 | 5 |
| 0 | 2 | 3 | 2 |
| 5 | 4 | 4 | 0 |
| 2 | 6 | 1 | 1 |
| 3 | 1 | 5 | 2 |
| 3 | 4 | 0 | 5 |
| 6 | 3 | 6 | 6 |
| 5 | 5 | 0 | 4 |
| 4 | 3 | 5 | 3 |
| 6 | 0 | 1 | 1 |
| 2 | 6 | 2 | 5 |
| 1 | 0 | 4 | 3 |
| 6 | 0 | 4 | 5 |
G:=sub<GL(4,GF(7))| [3,1,4,1,0,2,0,0,4,6,2,3,6,5,4,5],[0,5,2,3,2,4,6,1,3,4,1,5,2,0,1,2],[3,6,5,4,4,3,5,3,0,6,0,5,5,6,4,3],[6,2,1,6,0,6,0,0,1,2,4,4,1,5,3,5] >;
S3×SD32 in GAP, Magma, Sage, TeX
S_3\times {\rm SD}_{32} % in TeX
G:=Group("S3xSD32"); // GroupNames label
G:=SmallGroup(192,472);
// by ID
G=gap.SmallGroup(192,472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,135,184,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations