Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#857 |
| Fitting | C43 ⋊ D3 |
| derived | C42⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #857 |
| commutator, hypocenter | C42⋊C3 | C2×C4 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C43 | D3 |
| prime radical | C2×#857 | C2 |
| cosocle | C2×C42⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C4 | |
164 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,5,3,2)(4,10,6,11,7,12,9,8);(1,6,11,5,4,10,3,9,8,2,7,12)40 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#857, #11710998, #857⋊C8, … |
| maximal normal subgroup of | C8×#857, #11710998, #857⋊C8, … |
| maximal quotient of | C8×#857, #11710998, C4×#4622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5557);