Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8×C17 · Schur multiplier (H2) = C29 |
| direct product | C17×#62624 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C17 | C24 |
| commutator | C2 | C24×C8×C17 |
| Frattini, cosocle | C8 | C25×C17 |
| socle | C2×C17 | C24×C8 |
| prime radical | C17×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C17 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,27,23,28,37,47,24,29,38,48,39,49,59,69)(20,25,32,42,33,43,55,65,31,41,53,63,54,64,73,78)(21,26,35,45,36,46,58,68,34,44,56,66,57,67,74,79)(30,40,51,61,52,62,72,77,50,60,70,75,71,76,80,81);(18,20)(19,25)(21,30)(22,32)(23,33)(24,31)(26,40)(27,42)(28,43)(29,41)(34,50)(35,51)(36,52)(37,55)(38,53)(39,54)(44,60)(45,61)(46,62)(47,65)(48,63)(49,64)(56,70)(57,71)(58,72)(59,73)(66,75)(67,76)(68,77)(69,78)(74,80)(79,81);(18,21)(19,26)(20,30)(22,35)(23,36)(24,34)(25,40)(27,45)(28,46)(29,44)(31,50)(32,51)(33,52)(37,58)(38,56)(39,57)(41,60)(42,61)(43,62)(47,68)(48,66)(49,67)(53,70)(54,71)(55,72)(59,74)(63,75)(64,76)(65,77)(69,79)(73,80)(78,81);(21,34)(26,44)(30,50)(35,56)(36,57)(40,60)(45,66)(46,67)(51,70)(52,71)(58,74)(61,75)(62,76)(68,79)(72,80)(77,81);(20,31)(25,41)(30,50)(32,53)(33,54)(40,60)(42,63)(43,64)(51,70)(52,71)(55,73)(61,75)(62,76)(65,78)(72,80)(77,81)2312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 55612);