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 | D4×#41⋊C3 |
| unambiguous semidirect product | (D4×#41)⋊C3 |
| Fitting, Sylow | D4×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | D4×#41 | C3 |
| hypercenter | C4×D4 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
398 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 398 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) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,12)(7,11,15,18,8,17,14,20,13,10,16,19);(2,4)(5,13,9,15)(6,17,12,19)(7,16,14,8)(10,20,18,11);(1,2)(3,4)70 conjugacy classes — view the full character table.
| maximal subgroup of | #17507⋊C3, D4×#131⋊C3, #60146⋊C3, … |
| maximal normal subgroup of | #17507⋊C3, D4×#131⋊C3, #60146⋊C3, … |
| maximal quotient of | C22⋊C4×#41⋊C3, C4⋊C4×#41⋊C3, C4×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18173);