Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#33974 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C17 | C24 |
| commutator | C2 | C23×C16×C17 |
| Frattini, cosocle | C16 | C24×C17 |
| socle | C2×C17 | C23×C16 |
| prime radical | C2×#41×C17 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C17 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,27,22,28,34,47,23,29,35,48,37,49,56,69,24,26,36,44,38,45,57,66,39,46,58,67,59,68,74,79)(20,25,31,41,32,42,53,63,33,43,54,64,55,65,73,78,30,40,50,60,51,61,70,75,52,62,71,76,72,77,80,81);(18,20)(19,25)(21,31)(22,32)(23,33)(24,30)(26,40)(27,41)(28,42)(29,43)(34,53)(35,54)(36,50)(37,55)(38,51)(39,52)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(56,73)(57,70)(58,71)(59,72)(66,75)(67,76)(68,77)(69,78)(74,80)(79,81);(20,30)(25,40)(31,50)(32,51)(33,52)(41,60)(42,61)(43,62)(53,70)(54,71)(55,72)(63,75)(64,76)(65,77)(73,80)(78,81);(19,26)(25,40)(27,44)(28,45)(29,46)(41,60)(42,61)(43,62)(47,66)(48,67)(49,68)(63,75)(64,76)(65,77)(69,79)(78,81)2312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26962);