Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2804 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#390 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | #2804 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
480 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,11)(8,10)(20,22,23,27)(21,25,24,26);(5,9)(6,11)(12,14,15,18)(13,16,17,19);(4,6)(5,8)(7,11)(9,10)(14,18)(16,19)(21,24)(22,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,17)(14,16,18,19)(20,21,23,24)(22,26,27,25)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=eg,b^a=bef,c^a=cfg,d^a=deg,h^a=h^2,c^b=cg,d^b=def60 conjugacy classes — view the full character table.
| maximal subgroup of | #10683944, #10683947, #10684001, … |
| maximal normal subgroup of | #10683944, #10683947, #10684001, … |
| maximal quotient of | #10723604, #10723771, #10723950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8997);