Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#983 |
| Fitting | C23×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C23×Q8 | D3 |
| hypercenter | C24 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C24 | |
803 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 803 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) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,7,14)(6,17,19,10,11,20)(9,15,16)(12,18);(5,6)(7,12)(8,11)(9,10)(13,17)(14,20)(15,18)(16,19);(3,4);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11712723, #11712726, C2×#73074, … |
| maximal normal subgroup of | #11712723, #11712726, C2×#73074, … |
| maximal quotient of | C22×#5420, C22×#5394, C22×#5388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20056);