Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#578⋊C3 |
| unambiguous semidirect product | (C2×#578)⋊C3 |
| Fitting, Sylow | C2×#578 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×A4 |
| commutator, cosocle | #41 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C2×#578 | C3 |
| hypercenter | C2×D8 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
343 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2)(3,4)(5,14)(6,11,17,10,7,23)(8,12)(9,13)(15,30,31,29,16,34)(18,26,19,24,20,25)(21,28,33,27,22,32);(1,2)(3,17,8,19)(4,33,12,34)(5,32,13,31)(6,20,18,7)(9,23,14,25)(10,30,24,22)(11,21,26,29)(15,16,27,28);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #33382⋊C3, #33393⋊C3, C4×#578⋊C3, … |
| maximal normal subgroup of | #33382⋊C3, #33393⋊C3, C4×#578⋊C3, … |
| maximal quotient of | C2×#2898⋊C3, C2×#2913⋊C3, C4×#578⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18214);