Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #2656⋊C3 |
| Fitting, Sylow | #2656 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#127⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2656 | C3 |
| hypercenter | C22 | #127⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2 ◁ C22 | |
169 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 169 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:
(1,3,18,5,9,10)(2,14,20)(4,11,21,13,19,22)(6,7,15)(8,16,24)(12,17,23);(1,12,5,8)(2,4,6,13)(3,16,9,17)(7,19,14,11)(10,20,18,15)(21,23,22,24)Polycyclic presentation:
2^2,3,2^5|d^2=g,e^2=h,b^a=bf,c^a=cfh,d^a=dfg,e^a=efh,c^b=cgh,d^c=defh,e^c=dgh,g^c=fgh,h^c=g,e^d=ef24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2656⋊C3, #34641⋊C3, #35921⋊C3, … |
| maximal normal subgroup of | C2×#2656⋊C3, #34641⋊C3, #35921⋊C3, … |
| maximal quotient of | #14321⋊C3, #14322⋊C3, #14353⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5772);