Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | Q8×#41⋊C3 |
| Fitting, Sylow | Q8×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | Q8×#41 | C3 |
| prime radical | C2×#41⋊C3 | C22 |
| hypercenter | C4×Q8 | A4 |
| hypocenter | Q8 | C2×C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
268 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,13,16)(11,15,19,22,12,21,18,24,17,14,20,23);(1,3,4,8)(2,5,6,7)(9,17,13,19)(10,21,16,23)(11,20,18,12)(14,24,22,15);(1,2,4,6)(3,7,8,5)70 conjugacy classes — view the full character table.
| maximal subgroup of | #17509⋊C3, Q8×#131⋊C3, #60195⋊C3, … |
| maximal normal subgroup of | #17509⋊C3, Q8×#131⋊C3, #60195⋊C3, … |
| maximal quotient of | C4⋊C4×#41⋊C3, C4×#557⋊C3, C4×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18177);