Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#435 ⋊ C2 |
| Sylow | C3 ⋊ #3271 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#435 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #435 | D3 |
| hypocenter | C3 | #3271 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #435 | |
488 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)(8,11,13,19)(9,15,16,23)(10,17,12,21)(14,22,20,18);(5,7)(8,10,13,12)(9,14,16,20)(11,17,19,21)(15,22,23,18);(2,3)(8,9,13,16)(10,14,12,20)(11,18,19,22)(15,21,23,17);(2,3)(4,5)(6,7)(9,14)(10,12)(15,22)(16,20)(17,21)(18,23)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^b=de,e^b=eg,h^b=h^2,e^d=eg,h^d=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3271, C2×#79508, #11670260, … |
| maximal normal subgroup of | D3×#3271, C2×#79508, #11670260, … |
| maximal quotient of | #10784644, #10784764, #10784989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12084);