Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C4⋊(#41⋊C3) |
| unambiguous semidirect product | (C2×#550)⋊C3 |
| Fitting, Sylow | C2×#550 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×#550 | C3 |
| hypercenter | C2×C4⋊C4 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
411 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(3,4)(5,6)(7,9,12,8)(10,16,19,13,11,22,18,14,17,15,20,21);(1,2)(4,6)(7,21)(8,19)(9,17)(10,16)(11,13)(12,22)(14,18)(15,20);(1,2)70 conjugacy classes — view the full character table.
| maximal subgroup of | #17419⋊C3, #22536⋊C3, #22540⋊C3, … |
| maximal normal subgroup of | #17419⋊C3, #22536⋊C3, #22540⋊C3, … |
| maximal quotient of | C2×#2341⋊C3, C2×C4⋊C4×SL(2,𝔽3), C4×C4⋊(#41⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18172);