Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#465 ⋊ C2 |
| Sylow | C3 ⋊ #3062 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#465 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #465 | D3 |
| hypocenter | C3 | #3062 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #465 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(9,15)(10,16)(13,20)(17,22);(1,2,3)(5,7)(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23);(8,9,11,13,12,15,18,20)(10,14,17,19,16,21,22,23);(4,5)(6,7)(9,13)(11,18)(14,19)(15,20)(17,22)(21,23)Polycyclic presentation:
2^7,3|b^2=e,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg,d^c=dg,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3062, C2×#80661, #11662439, … |
| maximal normal subgroup of | D3×#3062, C2×#80661, #11662439, … |
| maximal quotient of | #10778664, #10778686, #10778708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13237);