Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting | C3×#397 ⋊ C2 |
| Sylow | C3 ⋊ #2736 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2736 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(6,11)(13,17)(16,19);(2,3)(13,17)(16,19)(22,26)(24,27);(4,6,7,11)(5,8,9,10)(12,14,15,18)(13,16,17,19)(20,22)(21,24)(23,26)(25,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,24,26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,b^a=beg,c^a=cfg,d^a=def,h^a=h^2,c^b=cg,d^b=deg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707804, #10707808, #10708288, … |
| maximal normal subgroup of | #10707804, #10707808, #10708288, … |
| maximal quotient of | #10722590, #10722592, #10724064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10282);