Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2809 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#389 | C2 |
| hypercenter | #389 | D3 |
| hypocenter | C3 | #2809 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #389 | |
368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,11)(8,10)(20,23,24,22)(21,25,27,26);(5,9)(6,11)(12,14,15,18)(13,16,17,19)(20,22,24,23)(21,26,27,25);(4,6)(5,8)(7,11)(9,10)(14,18)(16,19)(20,22,24,23)(21,25,27,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,17)(14,16,18,19)(20,21)(22,25)(23,26)(24,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=f,d^2=ef,b^a=beg,c^a=cfg,d^a=def,h^a=h^2,c^b=cg,d^b=deg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10683935, #10683938, #10684029, … |
| maximal normal subgroup of | #10683935, #10683938, #10684029, … |
| maximal quotient of | #10723656, #10723659, #10723883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8762);