Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12751 |
| 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 | |
876 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)(20,31,35,59)(21,37,40,55)(22,42)(23,28)(24,30)(25,33,49,65)(26,39,51,41)(27,44)(29,46)(32,60)(34,58,36,52)(38,70,73,50)(43,56,66,48)(45,53)(47,75)(54,64)(57,67)(61,76)(62,80)(63,79)(68,81)(69,72,78,74)(71,77);(18,20)(19,25)(21,32)(22,43)(23,34)(24,35)(26,47)(27,52)(28,48)(29,56)(30,49)(31,53)(33,64)(36,46)(37,68)(38,71)(39,62)(40,63)(41,67)(42,59)(44,65)(45,66)(50,60)(51,77)(54,58)(55,76)(57,78)(61,72)(69,80)(70,79)(73,75)(74,81);(18,21,24,40)(19,26,30,51)(20,32,35,63)(22,38,45,73)(23,41,46,39)(25,47,49,77)(27,50,54,70)(28,55,29,37)(31,57,59,80)(33,61,65,81)(34,67,36,62)(42,69,53,78)(43,71,66,75)(44,74,64,72)(48,76,56,68)(52,60,58,79)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5739);