Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×#117 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×#117 | D3 |
| hypocenter | C3 | #2346 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
504 conjugacy classes of subgroups; only the 233 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 233 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,19,20,18)(17,22,21,23);(9,13)(12,15)(16,18,20,19)(17,22,21,23);(8,10,11,14)(9,12,13,15)(16,17)(18,22)(19,23)(20,21);(2,3)(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(17,21)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de90 conjugacy classes — view the full character table.
| maximal subgroup of | #10692360, #10692364, #10692388, … |
| maximal normal subgroup of | #10692360, #10692364, #10692388, … |
| maximal quotient of | #10716612, #10716639, #10716669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14815);