Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #85646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | C22×#126 ⋊ C3 |
| derived | C22×C42 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C2×C3 |
| Frattini | C22 | C2×C22⋊A4 |
| Fitting | C22×#126 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(6,17,19,10,11,13)(7,20,18,15,16,14)(8,9,12);(1,4)(2,3)(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,17)(19,20);(5,13,9,20)(6,17,14,11)(7,10,15,18)(8,16,12,19)Polycyclic presentation:
2,3,2^6|c^2=gh,d^2=g,c^a=cg,d^a=dh,c^b=dfg,d^b=cdef,e^b=efgh,f^b=eg,g^b=h,h^b=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #8545⋊C3, #15986⋊C3, C2×#85646, … |
| maximal normal subgroup of | #8545⋊C3, #15986⋊C3, C2×#85646, … |
| maximal quotient of | #11709766, #11709851, #11709861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18222);