Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2442 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C22×C3×C4⋊C4 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C22×C4⋊C4 | D3 |
| hypocenter | C3 | #2442 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,19,20,18)(17,23,22,21);(2,3)(5,7)(8,10,11,15)(9,12,13,14)(17,22)(21,23);(4,5)(6,7)(16,18,20,19)(17,21,22,23);(8,9,11,13)(10,14,15,12)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10852268, #10852273, #10852343, … |
| maximal normal subgroup of | #10852268, #10852273, #10852343, … |
| maximal quotient of | #10717175, #10717439, #10718098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14748);