Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #95⋊C3 |
| automorphism | Aut(G) = C2×#4626⋊C2 (order 768) · Inn(G) = C42⋊C3 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #376⋊C3 |
| Fitting, Sylow | #376 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C42⋊C3 |
| commutator, cosocle, hypocenter | #95 | C2×C3 |
| Frattini, socle | C23 | C2×A4 |
| Fitting | #376 | C3 |
| prime radical | #95⋊C3 | C2 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,6,20,8,3,14,5,17,9,2,12,23)(4,16,19,18,11,24,13,15,21,7,10,22);(1,4,5,13)(2,7,8,18)(9,19)(10,11)(14,22)(15,16)(20,21)(23,24)Polycyclic presentation:
2,3,2^5|a^2=g,c^2=ef,d^2=e,c^b=d,d^b=cdf,e^b=fg,f^b=efg,d^c=dg24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 190);