Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2369 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | #2369 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
496 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 261 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) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,18,19,22)(17,21,20,23);(8,10,11,15)(9,12,13,14);(2,3)(4,6,7,5)(17,20)(18,22);(2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=fg,d^2=e,b^a=be,c^a=cfg,h^a=h^2,d^b=de,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10689644, #10689648, #10689651, … |
| maximal normal subgroup of | #10689644, #10689648, #10689651, … |
| maximal quotient of | #10716627, #10716737, #10716803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15806);