Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | #2840 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, cosocle | C25 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2840 | C3 |
| prime radical | C2×C22⋊A4 | C22 |
| hypocenter | C24 | C3×Q8 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,10,20,11,4,18,5,19,12,3,13,24)(2,14,22,9,7,21,8,17,15,6,16,23);(1,8)(2,5)(3,9)(4,7)(6,11)(10,17)(12,22,20,15)(13,16)(14,19)(18,23,24,21)Polycyclic presentation:
2^2,3,2^5|a^2=d,b^2=d,b^a=bd,e^a=eg,f^a=fgh,e^b=eh,f^b=fg,e^c=fh,f^c=efgh,g^c=gh,h^c=g22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73211, #11710041, #45458⋊C3, … |
| maximal normal subgroup of | C2×#73211, #11710041, #45458⋊C3, … |
| maximal quotient of | #14673⋊C3, #17308⋊C3, #11709342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5787);