Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #2615⋊C3 |
| Fitting, Sylow | #2615 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2615 | C3 |
| hypercenter | C4 | #120⋊C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
197 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,7,20,5,3,17,6,15,9,2,12,24)(4,18,22,16,14,19,13,10,23,8,11,21);(1,16,6,8)(2,13,5,4)(3,7)(9,23)(10,11)(12,15)(14,18)(17,21)(19,24)(20,22)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=h,e^2=h,b^a=bh,d^a=dfh,e^a=egh,d^c=eh,e^c=de,f^c=g,g^c=fgh,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | #32715⋊C3, #32716⋊C3, #32884⋊C3, … |
| maximal normal subgroup of | #32715⋊C3, #32716⋊C3, #32884⋊C3, … |
| maximal quotient of | #14301⋊C3, #14337⋊C3, #14885⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5767);