Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2722 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2722 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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)(8,10,11,14)(9,13,12,15)(17,22)(21,23);(2,3)(5,7)(9,12)(13,15)(16,19,20,18)(17,21,22,23);(2,3)(4,5)(6,7)(9,12)(10,14)(16,18,20,19)(17,21,22,23);(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^7,3|b^2=ef,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^2,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707580, #10707584, #10708166, … |
| maximal normal subgroup of | #10707580, #10707584, #10708166, … |
| maximal quotient of | #10722624, #10722915, #10723014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10168);