Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C24×C4 |
| direct product | C11×#22285 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #2328 |
| commutator | C2×C4 | C23×C4×C11 |
| Frattini, cosocle | C22×C4 | C24×C11 |
| prime radical | C2×C11×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,25)(15,31,35,45)(16,36)(17,23)(18,24)(19,28)(20,34,46,60)(21,38)(22,37)(26,51,54,40)(27,55)(29,42)(30,49)(32,43,53,68)(33,44,63,69)(39,47)(41,58)(48,65)(50,56)(52,72,74,67)(57,70)(59,75,64,62)(61,71,73,66);(12,14,17,28,18,30,37,29)(13,19,22,25,24,42,23,49)(15,32,60,26,35,53,34,54)(16,27,38,56,39,58,65,57)(20,43,31,40,46,68,45,51)(21,41,36,50,48,55,47,70)(33,61,75,52,63,73,62,74)(44,67,59,66,69,72,64,71);(12,15,18,35)(13,20)(14,26)(16,33,39,63)(17,34,37,60)(19,40,42,51)(21,44)(22,45)(23,31)(24,46)(25,43,49,68)(27,52)(28,32)(29,53)(30,54)(36,64)(38,62,65,75)(41,66,55,71)(47,59)(48,69)(50,67,70,72)(56,61)(57,73)(58,74);(12,16,37,65,18,39,17,38)(13,21,23,47,24,48,22,36)(14,27,29,57,30,58,28,56)(15,33,60,75,35,63,34,62)(19,41,49,70,42,55,25,50)(20,44,31,59,46,69,45,64)(26,52,53,73,54,74,32,61)(40,66,68,72,51,71,43,67)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 15273);