Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting, derived, Sylow | #2840 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #2840 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6)(3,8,20)(4,12,15)(5,9,17)(7,16,14)(10,18,24)(11,21,23)(13,19,22);(1,10,5,14)(2,19,9,12)(3,13,11,4)(6,23,17,20)(7,21,18,8)(15,24,22,16);(3,11)(4,13)(7,18)(12,19)(16,24)(20,23)Polycyclic presentation:
3,2^7|c^2=h,d^2=h,b^a=be,c^a=d,d^a=cdh,e^a=b,f^a=g,g^a=fg,c^b=cf,d^b=dfg,d^c=dh,e^c=eg,e^d=ef14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5868);