Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#919 |
| Fitting | C22×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ C22×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D3×D4 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C22×C3×Q16 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | C22×Q16 | D3 |
| hypocenter | C3 | C22×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×Q16 | |
1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 525 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 — 5 generators on 23 points:
(1,2)(3,4)(5,6,7)(8,10,12,16)(9,14,13,22)(11,20,21,18)(15,23,19,17);(1,2)(3,4)(8,9,12,13)(10,17,16,23)(11,19,21,15)(14,20,22,18);(6,7)(9,13)(10,16)(15,19)(18,20);(3,4);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10710837, #10710858, #10796533, … |
| maximal normal subgroup of | #10710837, #10710858, #10796533, … |
| maximal quotient of | C22×#4804, C22×#4880, C22×#4872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19755);