Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#419 ⋊ C2 |
| Sylow | C3 ⋊ #3005 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#419 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #419 | D3 |
| hypocenter | C3 | #3005 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #419 | |
758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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)(9,16)(14,23)(15,20)(17,22);(2,3)(5,7)(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17);(2,3)(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20);(2,3)(4,5)(6,7)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23)Polycyclic presentation:
2^7,3|c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=h^2,e^b=eg,h^b=h^2,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10826550, #10826553, #10826587, … |
| maximal normal subgroup of | #10826550, #10826553, #10826587, … |
| maximal quotient of | #10767576, #10767615, #10767654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11535);