Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C11 |
| Frattini, cosocle | C22×C4 | C24×C11 |
| prime radical | C2×C11×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
1294 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)(14,24)(15,29,32,20)(16,21)(17,36)(18,23)(19,27)(22,39)(25,52,55,50)(26,51)(28,48)(30,59,63,43)(31,44,66,60)(33,67)(34,49)(35,47)(37,42)(38,46)(40,54,58,64)(41,57)(45,56)(53,69,73,72)(61,68,74,75)(62,70,65,71);(12,14,18,28)(13,19,23,42)(15,25,32,55)(16,33,35,57)(17,37,39,27)(20,40,29,58)(21,45,47,51)(22,48,36,24)(26,38,56,34)(30,61,63,74)(31,64,66,54)(41,49,67,46)(43,69,59,72)(44,52,60,50)(53,65,73,62)(68,71,75,70);(12,15)(13,20)(14,25)(16,30)(17,31)(18,32)(19,40)(21,43)(22,44)(23,29)(24,50)(26,53)(27,54)(28,55)(33,61)(34,62)(35,63)(36,60)(37,64)(38,65)(39,66)(41,68)(42,58)(45,69)(46,70)(47,59)(48,52)(49,71)(51,72)(56,73)(57,74)(67,75);(12,16,18,35)(13,21,23,47)(14,26,28,56)(15,30,32,63)(17,38,39,34)(19,41,42,67)(20,43,29,59)(22,49,36,46)(24,51,48,45)(25,53,55,73)(27,57,37,33)(31,65,66,62)(40,68,58,75)(44,71,60,70)(50,72,52,69)(54,74,64,61)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 15445);