Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4625 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#4625 |
| derived | C22⋊A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625 |
| commutator, hypocenter | C22⋊A4 | C2×C4 |
| Frattini | C23 | C2×S4 |
| Fitting | C2×#120 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C2×#120⋊C3 | C2 |
| cosocle | #120⋊C3 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
141 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(3,8,18,9,7,11,10,5)(4,17,13,15,12,14,16,6);(1,2)(4,9,18)(5,14,11)(6,7,15)(8,16,10)(12,13,17)22 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4625, #11711500, #5450⋊C4, … |
| maximal normal subgroup of | C4×#4625, #11711500, #5450⋊C4, … |
| maximal quotient of | C2×#67998, #11711340, #11711342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5655);