Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #2639 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C2×#930 | C2 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2639 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
576 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 149 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)(18,22)(21,23);(2,3)(4,5)(6,7)(9,13)(12,15)(16,18)(17,21)(19,22)(20,23);(2,3)(4,5,6,7)(8,10,11,14)(9,12,13,15)(17,20)(21,23);(2,3)(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,b^a=be,c^a=cf,h^a=h^2,c^b=cg,h^b=h^2,d^c=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10695721, #10695723, #10695733, … |
| maximal normal subgroup of | #10695721, #10695723, #10695733, … |
| maximal quotient of | #10721578, #10721706, #10721782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15400);