Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#5420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C23×Q8 | D3 |
| prime radical | C23×SL(2,𝔽3) | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C24 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C24 | |
355 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 355 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) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(7,13,16,8,9,21)(10,14)(11,19,18,12,17,22)(15,20);(3,4,5,6)(8,12)(9,10)(13,20)(14,19)(15,17)(16,18);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711309, #11711317, C4×#5420, … |
| maximal normal subgroup of | #11711309, #11711317, C4×#5420, … |
| maximal quotient of | C2×#73056, C2×#73057, #11711796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18079);