Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2579 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#386 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2579 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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:
(1,2,3)(9,13)(12,14)(17,20)(21,23);(2,3)(5,7)(16,18,19,22)(17,21,20,23);(2,3)(8,10,11,15)(9,12,13,14)(17,20)(18,22);(4,5)(6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|b^2=eg,c^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^2,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10713837, #10713850, #10849386, … |
| maximal normal subgroup of | #10713837, #10713850, #10849386, … |
| maximal quotient of | #10717581, #10717905, #10718554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7346);