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 ⋊ #3269 |
| 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 | #3269 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #435 | |
478 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)(8,12)(9,11)(10,15)(13,17)(14,16)(18,21)(19,22)(20,23);(5,7)(8,11,14,13)(9,16,17,12)(10,19,20,21)(15,22,23,18);(2,3)(8,10,14,20)(9,15,17,23)(11,19,13,21)(12,18,16,22);(2,3)(4,5)(6,7)(8,9,14,17)(10,18,20,22)(11,16,13,12)(15,21,23,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^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×#3269, C2×#79498, #11670262, … |
| maximal normal subgroup of | D3×#3269, C2×#79498, #11670262, … |
| maximal quotient of | #10784614, #10784734, #10784884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12074);