Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2022 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting, prime radical | C3×#534 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #534 | D3 |
| hypocenter | C3 | #2022 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #534 | |
430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 23 points:
(1,2,3)(5,7)(8,11)(9,14)(10,16)(12,18)(13,19)(15,22)(17,23)(20,21);(2,3)(8,10,12,17)(9,13,15,20)(11,16,18,23)(14,21,22,19);(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)(16,19)(17,20)(18,22)(21,23)Polycyclic presentation:
2^7,3|b^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,h^b=h^2,d^c=dg48 conjugacy classes — view the full character table.
| maximal subgroup of | #10707224, #10707226, #10707238, … |
| maximal normal subgroup of | #10707224, #10707226, #10707238, … |
| maximal quotient of | #10695409, #10695416, #10695423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4425);