Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting | C3×#391 ⋊ C2 |
| Sylow | C3 ⋊ #2832 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#391 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | #2832 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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)(6,10)(8,11)(20,23,24,27)(21,25,22,26);(5,9)(8,11)(14,18)(16,19)(20,22,24,21)(23,25,27,26);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,18)(17,19)(20,21)(22,24)(23,25)(26,27);(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(21,22)(25,26)Polycyclic presentation:
2^7,3|a^2=e,c^2=eg,d^2=eg,b^a=beg,c^a=cf,c^b=cg,d^b=defg,h^b=h^2,d^c=deg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10707882, #10707886, #10708366, … |
| maximal normal subgroup of | #10707882, #10707886, #10708366, … |
| maximal quotient of | #10725868, #10725945, #10725994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9237);