Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#427 ⋊ C2 |
| Sylow | C3 ⋊ #3058 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#427 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #427 | D3 |
| hypocenter | C3 | #3058 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 524 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,17)(15,22)(18,21)(19,23);(2,3)(5,7)(8,10)(9,15)(11,18)(12,17)(13,19)(14,21)(16,22)(20,23);(8,9,14,20,12,16,11,13)(10,15,21,23,17,22,18,19);(2,3)(4,5)(6,7)(9,13)(11,14)(15,19)(16,20)(18,21)(22,23)Polycyclic presentation:
2^7,3|b^2=eg,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^c=dg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10712635, #10712660, #10713919, … |
| maximal normal subgroup of | #10712635, #10712660, #10713919, … |
| maximal quotient of | #10778259, #10778298, #10778346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11759);