Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2637 |
| 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 | #2637 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 450 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)(16,19,20,18)(17,21,22,23);(5,7)(9,12)(13,15)(18,19)(21,23);(2,3)(4,5)(6,7)(16,18,20,19)(17,21,22,23);(8,9)(10,13)(11,12)(14,15)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10768629, #10768633, #10768659, … |
| maximal normal subgroup of | #10768629, #10768633, #10768659, … |
| maximal quotient of | #10721667, #10721699, #10721858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10120);