Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#451 ⋊ C2 |
| Sylow | C3 ⋊ #3013 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#451 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #451 | D3 |
| hypocenter | C3 | #3013 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #451 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(8,10,12,16)(9,14,15,21)(11,17,18,22)(13,20,19,23);(5,7)(10,16)(14,21)(17,22)(20,23);(8,9,12,15)(10,14,16,21)(11,19,18,13)(17,23,22,20);(2,3)(4,5)(6,7)(9,13)(11,18)(14,20)(15,19)(17,22)(21,23)Polycyclic presentation:
2^7,3|b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,e^b=eg,d^c=dg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3013, C2×#80186, #11661650, … |
| maximal normal subgroup of | D3×#3013, C2×#80186, #11661650, … |
| maximal quotient of | #10768403, #10768417, #10768491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12762);