Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C2×C13×#1403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C13 |
| Frattini, cosocle | #124 | C23×C13 |
| prime radical | C2×C13×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C23×C13 ◁ C22×C4×C13 ◁ C2×C13×#124 ◁ G | |
514 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4,6,10,8,5,12,14)(7,15,17,13,18,11,16,9);(3,5)(4,8)(6,13)(7,11)(9,12)(10,16)(14,17)(15,18)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6551);