Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| prime radical | C22×#41⋊C3 | C2 |
| hypercenter | C22×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
743 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,16,22,18,8,32,10,33,20,7,21,34)(6,23,28,15,12,35,14,29,26,11,27,36)(9,17,19,31)(13,24,25,30);(5,6,10,14)(7,15,18,11)(8,13,21,25)(9,12,19,27)(16,30,33,24)(17,29,31,23)(20,28,22,26)(32,35,34,36);(3,4);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11712724, #11712727, C2×#73077, … |
| maximal normal subgroup of | #11712724, #11712727, C2×#73077, … |
| maximal quotient of | C22×#5426, C22×#5388, C2×C4×#984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20060);