Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | Aut(G) = C2×C4×S4 (order 192) · Inn(G) = A4 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #504⋊C3 |
| Fitting, Sylow | #504 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | A4 |
| commutator, hypocenter | Q8 | C3×C8 |
| Frattini | C8 | C2×A4 |
| Fitting | #504 | C3 |
| socle | C2 | C8×A4 |
| prime radical | #41⋊C3 | C4 |
| cosocle | #131 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C16 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,8,4,9,13,24,7,12,16,27,19,30,37,48)(5,11,38,25,18,53,35,47,22,32,41,51,43,56,59,62,6,31,14,26,42,28,36,61,21,34,40,50,45,55,58,64,20,10,15,49,17,29,57,46,23,33,39,52,44,54,60,63);(1,20,7,22)(2,31,12,33)(3,38,16,40)(4,42,19,44)(5,23,21,6)(8,49,27,51)(9,53,30,55)(10,34,32,11)(13,57,37,59)(14,41,39,15)(17,45,43,18)(24,61,48,63)(25,52,50,26)(28,56,54,29)(35,60,58,36)(46,64,62,47)Polycyclic presentation:
2,3,2^5|a^2=c,c^2=d,d^2=g,e^2=g,f^2=g,e^b=f,f^b=ef,f^e=fg56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 204);