Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72214076 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2403 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C3 |
| socle | C22×C3 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(7,11)(9,16)(12,18)(13,17)(14,20)(19,21);(6,7)(8,12)(9,17)(10,13)(11,16)(14,21)(15,19)(18,20);(1,2)(6,8,10,15)(7,12,13,19)(9,14,16,20)(11,18,17,21)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1461);