Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#11871 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #1519 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C2×M16 | C23×C11 |
| prime radical | C2×C11×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C11×#131 ◁ G | |
360 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 110 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)(12,13,17,23,19,24,18,43)(14,25,28,51,30,56,29,64)(15,31,34,50,36,66,35,21)(16,37,40,22,42,53,41,54)(20,38,45,63,47,62,46,27)(26,57,32,67,60,74,59,55)(33,52,69,65,39,49,70,71)(44,58,72,75,73,61,48,68);(12,14,19,30)(13,20,24,47)(15,32,36,59)(16,38,42,62)(17,29,18,28)(21,48,50,72)(22,51,54,64)(23,45,43,46)(25,53,56,37)(26,35,60,34)(27,41,63,40)(31,44,66,73)(33,58,39,61)(49,55,52,67)(57,65,74,71)(68,70,75,69);(12,15,18,35,19,36,17,34)(13,21,23,31,24,50,43,66)(14,26,29,59,30,60,28,32)(16,39,40,70,42,33,41,69)(20,44,46,48,47,73,45,72)(22,52,37,71,54,49,53,65)(25,55,51,57,56,67,64,74)(27,61,38,68,63,58,62,75)374 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 4859);