Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2864 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2864 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
400 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 157 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)(5,9)(8,11)(20,22,23,26)(21,25,24,27);(2,3)(12,14,15,19)(13,16,17,18)(21,24)(25,27);(4,6,7,10)(5,8,9,11)(21,24)(25,27);(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,18,19,16)(20,21,23,24)(22,25,26,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10695284, #10695286, #10695289, … |
| maximal normal subgroup of | #10695284, #10695286, #10695289, … |
| maximal quotient of | #10726385, #10726445, #10726451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6798);