Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1786 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C2×C17×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,23,30,25,29,24,46)(20,31,34,59,35,58,47,55)(21,36,38,54,39,27,45,53)(22,40,42,56,44,73,43,28)(26,41,50,65,49,74,51,33)(32,60,62,79,64,57,63,78)(37,68,70,67,71,72,75,52)(48,69,76,61,66,80,77,81);(18,20)(19,26)(21,32)(22,41)(23,34)(24,47)(25,35)(27,48)(28,55)(29,49)(30,50)(31,40)(33,43)(36,66)(37,69)(38,62)(39,64)(42,65)(44,74)(45,63)(46,51)(52,78)(53,76)(54,77)(56,59)(57,72)(58,73)(60,68)(61,70)(67,79)(71,80)(75,81);(18,21,25,39)(19,27,29,36)(20,32,35,64)(22,37,44,71)(23,45,24,38)(26,48,49,66)(28,52,56,67)(30,54,46,53)(31,57,58,60)(33,61,65,81)(34,63,47,62)(40,72,73,68)(41,69,74,80)(42,75,43,70)(50,77,51,76)(55,78,59,79)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5732);