Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#376 ⋊ C2 |
| Sylow | C3 ⋊ #2644 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#376 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #376 | D3 |
| hypocenter | C3 | #2644 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #376 | |
420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 420 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(20,22,23,26)(21,25,24,27);(2,3)(5,9)(8,11)(14,18)(16,19)(22,26)(25,27);(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,18)(17,19)(21,24)(25,27);(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,16,18,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=fg,d^2=e,b^a=be,c^a=cfg,c^b=cg,d^c=de,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10898320, #10898326, #10898335, … |
| maximal normal subgroup of | #10898320, #10898326, #10898335, … |
| maximal quotient of | #10721580, #10721630, #10722301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6006);