Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#396 ⋊ C2 |
| Sylow | C3 ⋊ #2820 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2820 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(10,14)(12,15)(16,19,20,18)(17,21,22,23);(1,2,3)(5,7)(9,13)(12,15)(17,22)(18,19);(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,20,19)(17,21,22,23);(8,9,11,13)(10,12,14,15)(16,17)(18,23)(19,21)(20,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,d^2=ef,b^a=bef,c^a=cf,c^b=cg,d^b=de,d^c=def,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2820, C2×#77402, #11001736, … |
| maximal normal subgroup of | D3×#2820, C2×#77402, #11001736, … |
| maximal quotient of | #10725855, #10725858, #10725877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9978);