Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#7387 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | D16 |
| commutator | C16 | C2×C8×C11 |
| Frattini, cosocle | C4×C16 | C22×C11 |
| socle | C2×C11 | #1331 |
| prime radical | C11×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ C4×C11×C16 ◁ G | |
120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 120 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) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,22,18,23,38,51,19,25,39,52,42,24,17,40)(14,26,28,54,29,55,60,74,31,56,61,75,30,62,41,47)(15,32,34,48,36,63,69,50,37,65,70,49,71,64,35,21)(20,33,43,57,45,66,72,59,46,68,73,58,53,67,44,27);(12,14)(13,20)(15,33)(16,28)(17,41)(18,29)(19,31)(21,47)(22,43)(23,45)(24,53)(25,46)(26,32)(27,35)(30,42)(34,57)(36,66)(37,68)(38,60)(39,61)(40,44)(48,54)(49,75)(50,74)(51,72)(52,73)(55,63)(56,65)(58,70)(59,69)(62,64)(67,71)836 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 375);