Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #338⋊C3 |
| automorphism | Aut(G) = #4626⋊C2 (order 384) · Inn(G) = C42⋊C3 · Out(G) = D4 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #338⋊C3 |
| Fitting, derived, Sylow | #338 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C42⋊C3 |
| commutator, Fitting, cosocle, hypocenter | #338 | C3 |
| Frattini | C22×C4 | A4 |
| socle | C2 | #95⋊C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C4 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,7)(8,13,10)(9,12,15)(11,16,14);(1,15,5,16,6,7,4,10)(2,11,14,12,9,13,8,3)Polycyclic presentation:
3,2^6|b^2=de,c^2=d,d^2=g,f^2=g,b^a=c,c^a=bce,d^a=efg,e^a=def,c^b=cf,d^b=dg,e^b=eg,e^c=eg18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 4);