Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#388 ⋊ C2 |
| Sylow | C3 ⋊ #2337 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#388 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×#67 | C22 |
| hypercenter | #388 | D3 |
| hypocenter | C3 | #2337 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #388 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(9,12)(13,14)(16,18,19,22)(17,21,20,23);(1,2,3)(8,10,11,15)(9,13,12,14)(17,20)(21,23);(4,6,7,5)(9,12)(10,15);(4,5,7,6)(8,9,11,12)(10,14,15,13)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,c^b=ce,h^d=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2337, C2×#75472, #10954110, … |
| maximal normal subgroup of | D3×#2337, C2×#75472, #10954110, … |
| maximal quotient of | #10716458, #10716515, #10716521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8048);