Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ #2427 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#124 | C2 |
| prime radical | C2×#938 | C2 |
| hypercenter | C2×#124 | D3 |
| hypocenter | C3 | #2427 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,14,15,19)(13,17,16,18);(2,3)(5,7)(8,9)(10,11);(2,3)(4,5)(6,7)(8,9,10,11)(13,16)(17,18);(2,3)(8,9,10,11)(12,13,15,16)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,d^2=ef,b^a=bef,c^a=cf,d^a=def,h^a=h^2,c^b=cg,h^b=h^2,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10842593, #10842596, #10842668, … |
| maximal normal subgroup of | #10842593, #10842596, #10842668, … |
| maximal quotient of | #10717130, #10717558, #10717995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15572);