Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#576⋊C3 |
| unambiguous semidirect product | (C2×#576)⋊C3 |
| Fitting, Sylow | C2×#576 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×A4 |
| commutator, cosocle | #41 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C2×#576 | C3 |
| hypercenter | C2×Q16 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×Q16 | |
347 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 347 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) = 50 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 50 points:
(1,2)(3,15,39,18,6,31,10,35,19,5,23,47)(4,27,38,28,12,43,13,33,29,11,21,46)(7,41,37,16,20,50,9,32,36,25,26,45)(8,34,49,14,22,48,24,30,42,17,40,44);(1,2)(5,16)(6,22)(7,9)(8,24)(11,14)(12,20)(15,27)(17,28)(18,25)(19,36)(21,26)(23,40)(29,49)(30,41)(32,34)(33,35)(37,39)(38,42)(43,46)(45,50);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #33381⋊C3, #33392⋊C3, C4×#576⋊C3, … |
| maximal normal subgroup of | #33381⋊C3, #33392⋊C3, C4×#576⋊C3, … |
| maximal quotient of | C2×#2897⋊C3, C2×#2913⋊C3, C4×#576⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18212);