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×#12742 |
| 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 | |
878 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)(20,30)(21,35,37,26)(22,39)(23,44)(24,28)(25,32)(27,42)(29,41)(31,57,59,55)(33,63)(34,56)(36,67,70,73)(38,50,69,66)(40,53)(43,54)(45,51)(46,58,64,80)(47,62)(48,74)(49,71,65,79)(52,61)(60,76,72,78)(68,75,81,77);(18,20,24,34)(19,25,28,47)(21,31,37,59)(22,40,43,74)(23,45,41,33)(26,46,35,64)(27,51,52,63)(29,53,44,48)(30,54,56,39)(32,61,62,42)(36,68,70,81)(38,72,69,60)(49,76,65,78)(50,77,66,75)(55,67,57,73)(58,79,80,71);(18,21)(19,26)(20,31)(22,36)(23,38)(24,37)(25,46)(27,49)(28,35)(29,50)(30,55)(32,58)(33,60)(34,59)(39,73)(40,68)(41,69)(42,71)(43,70)(44,66)(45,72)(47,64)(48,75)(51,76)(52,65)(53,77)(54,67)(56,57)(61,79)(62,80)(63,78)(74,81)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5730);