Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5923 |
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| socle | C22 | C22×S4 |
| hypercenter | C22×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 | |
757 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 757 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 4 generators on 34 points:
(1,2)(3,6,17)(4,23,25,10,11,26)(5,13,28)(8,18,19)(9,31,33,20,21,34)(12,24)(15,29,30)(22,32);(3,5,8,15)(4,9,10,20)(6,13,18,29)(7,14,16,27)(11,21,23,31)(12,22,24,32)(17,28,19,30)(25,33,26,34);(3,4)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,20)(16,23)(17,26)(18,24)(19,25)(27,31)(28,34)(29,32)(30,33);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11712717, #11712718, C2×#73132, … |
| maximal normal subgroup of | #11712717, #11712718, C2×#73132, … |
| maximal quotient of | C2×#85519, C2×#85341, C2×C4×#986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20064);