Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2193 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#131 ◁ G | |
210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(2,8)(3,12)(4,15)(5,6)(9,23)(10,27)(11,22)(13,29)(14,19)(16,20)(17,18)(21,24)(25,32)(26,30)(33,34,35);(1,2)(3,9)(4,16)(5,10)(6,20)(7,11)(8,15)(12,24)(13,30)(14,26)(17,22)(18,27)(19,25)(21,28)(23,29)(31,32);(1,3,7,14)(2,9,11,26)(4,13,18,29)(5,19,6,12)(8,21,22,32)(10,25,20,24)(15,28,17,31)(16,30,27,23)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5484);