Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C3 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G | |
236 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(3,9)(4,7)(5,13)(6,10)(8,12)(11,16)(14,15)(17,18,19);(2,7)(3,5)(4,12)(8,10)(9,15)(13,16);(1,3)(4,11)(5,6)(7,8)(9,13)(12,14)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5495);