Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2604 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2604 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
470 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(9,13)(12,14)(16,18,19,22)(17,21,20,23);(4,5,6,7)(17,20)(21,23);(8,10,11,15)(9,12,13,14)(16,17)(18,21)(19,20)(22,23);(2,3)(5,7)(8,9,11,13)(10,14,15,12)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10768981, #10768985, #10769011, … |
| maximal normal subgroup of | #10768981, #10768985, #10769011, … |
| maximal quotient of | #10717844, #10717910, #10719354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10621);