Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #2663 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2663 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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)(10,14)(13,15)(16,18,19,22)(17,21,20,23);(2,3)(5,7)(8,10)(9,13)(11,14)(12,15)(17,20)(21,23);(2,3)(4,5,6,7)(9,12)(10,14);(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|b^2=eg,d^2=f,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10770128, #10770132, #10770158, … |
| maximal normal subgroup of | #10770128, #10770132, #10770158, … |
| maximal quotient of | #10721758, #10721771, #10721836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15401);