Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#5404 |
| Fitting | C2×#131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×#131 | D3 |
| socle | C22 | C4×S4 |
| prime radical | C2×#985 | C2 |
| cosocle | #41⋊C3 | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C8 | |
263 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(3,10,28,9,18,26,15,33,17,4,13,32,8,25,30,22,6,24,5,20,19,12,29,34)(7,11,14,21,16,23,27,31);(6,7)(10,11)(13,14)(16,18)(17,19)(20,21)(23,25)(24,26)(27,29)(28,30)(31,33)(32,34);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | #11711561, C4×#5404, #11712583, … |
| maximal normal subgroup of | #11711561, C4×#5404, #11712583, … |
| maximal quotient of | C2×C8×GL(2,𝔽3), C2×C8×#226, C8×#985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18022);