Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#7388 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #357 |
| commutator | C16 | C2×C8×C13 |
| Frattini, cosocle | C4×C16 | C22×C13 |
| socle | C2×C13 | #1331 |
| prime radical | C13×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C8×C13 ◁ C4×C8×C13 ◁ C4×C13×C16 ◁ G | |
118 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,20,25,42,52,21,27,37,57,45,26,19,43)(16,28,30,60,32,62,44,61,34,63,31,69,33,70,65,51)(17,35,38,53,40,71,64,55,41,59,75,54,76,72,39,23)(22,46,47,67,49,74,48,29,50,36,56,66,58,73,77,68);(14,16)(15,22)(17,36)(18,31)(19,44)(20,32)(21,34)(23,51)(24,56)(25,49)(26,58)(27,50)(28,59)(29,64)(30,37)(33,45)(35,63)(38,67)(39,68)(40,73)(41,46)(42,65)(43,48)(47,57)(52,77)(53,60)(54,69)(55,61)(62,72)(66,75)(70,71)(74,76)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 376);