Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2814 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#394 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #394 | D3 |
| hypocenter | C3 | #2814 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #394 | |
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:
(1,2,3)(10,14)(12,15)(16,18)(17,21)(19,23)(20,22);(5,7)(9,13)(12,15)(18,23)(20,21);(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,19,23)(17,20,22,21);(2,3)(8,9)(10,12)(11,13)(14,15)(16,17,19,22)(18,21,23,20)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^b=def,d^c=de48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2814, C2×#77084, #11001352, … |
| maximal normal subgroup of | D3×#2814, C2×#77084, #11001352, … |
| maximal quotient of | #10725836, #10725840, #10725866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9660);