Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C2×#72216366 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22⋊C4 |
| commutator | C22 | C22×C3×C4 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C23×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 845);