Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#1007 |
| Fitting | C23×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ C22×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C23×C3×Q8 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | C22×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
5872 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3221 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) = 192 |
A minimal faithful permutation representation — 6 generators on 23 points:
(1,2)(3,4)(5,6,7)(9,15)(13,21)(14,23)(20,22);(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,20,21,22);(8,10,12,17)(9,13,15,21)(11,18,19,16)(14,22,23,20);(6,7)(8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23);(3,4);(1,2)108 conjugacy classes — view the full character table.
| maximal subgroup of | #10794331, #10794358, #10794359, … |
| maximal normal subgroup of | #10794331, #10794358, #10794359, … |
| maximal quotient of | C22×#5572, C22×#5812, C22×#5504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20137);