Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #85645 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | C22×#120 ⋊ C3 |
| derived | C26 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C26 | C2×C3 |
| Frattini | C22 | C2×C22⋊A4 |
| Fitting | C22×#120 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4,3)(5,6)(7,9,11,10,8,12);(1,3)(2,4);(6,12)(9,10)Polycyclic presentation:
2,3,2^6|c^a=cg,d^a=dh,c^b=cdeh,d^b=cef,e^b=efg,f^b=egh,g^b=gh,h^b=g24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18221);