Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#13199 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | #124 |
| commutator | C22×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C2×C11 | #1955 |
| prime radical | C11×#575 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C22×C8×C11 ◁ G | |
406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,23,18,24,39,53,19,25,40,54,16,22,27,42)(14,26,31,55,29,44,56,52,30,50,38,66,36,20,41,45)(15,21,34,48,33,47,62,71,35,49,65,74,37,51,57,68)(28,58,61,73,32,43,63,70,60,46,67,72,59,69,64,75);(12,14,27,56,16,36,40,31,19,30,39,41,18,29,17,38)(13,20,42,55,22,50,54,45,25,44,53,66,24,26,23,52)(15,32,62,61,37,28,34,64,35,59,57,67,33,60,65,63)(21,46,71,70,51,43,48,73,49,58,68,75,47,69,74,72);(12,15)(13,21)(14,28,30,60)(16,37)(17,34)(18,33)(19,35)(20,43,44,69)(22,51)(23,48)(24,47)(25,49)(26,46,50,58)(27,57)(29,32,36,59)(31,61,38,67)(39,62)(40,65)(41,64,56,63)(42,68)(45,70,52,75)(53,71)(54,74)(55,72,66,73)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6187);