Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = C2×#72218858 (order 46080) · Inn(G) = C24 · Out(G) = C22×S6 |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C3 |
| socle | C2×C3 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,19);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1536);