Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#391 | C2 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | #2568 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
468 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,19,20,18)(17,23,22,21);(4,5,6,7)(10,14)(12,15)(17,22)(21,23);(2,3)(8,10,11,14)(9,12,13,15)(16,18,20,19)(17,21,22,23);(2,3)(5,7)(8,9,11,13)(10,12,14,15)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=efg,d^2=eg,b^a=beg,c^a=cf,d^a=deg,h^a=h^2,c^b=cg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10713868, #10829777, #10829781, … |
| maximal normal subgroup of | #10713868, #10829777, #10829781, … |
| maximal quotient of | #10717472, #10717600, #10718080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9120);