Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72214271 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2403 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#103 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | D4 |
| commutator | C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C4 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(3,4,5);(6,7,8,9)(11,14)(12,16)(15,17);(6,7)(8,9)(10,11,13,15)(12,17,16,14)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 848);