Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13096 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1898 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×#41 | C23×C17 |
| prime radical | C2×C17×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C17×2−1+4 ◁ G | |
624 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)(20,30,35,59)(21,36,39,55)(22,40,31,27)(23,28)(24,29)(25,33)(26,38,51,45)(32,61)(34,58,44,75)(37,70)(41,49)(42,69)(43,72,60,53)(46,56)(47,76,79,81)(48,66)(50,71)(52,74)(54,73)(57,67)(62,68,65,78)(63,77)(64,80);(18,20,31,33,24,35,22,41)(19,25,40,59,29,49,27,30)(21,32,37,65,39,64,42,62)(23,44,60,73,46,34,43,66)(26,47,50,77,51,79,52,57)(28,54,72,58,56,48,53,75)(36,68,69,61,55,78,70,80)(38,67,71,81,45,63,74,76);(18,21)(19,26)(20,32)(22,42)(23,45)(24,39)(25,47)(27,52)(28,55)(29,51)(30,57)(31,37)(33,65)(34,67)(35,64)(36,56)(38,46)(40,50)(41,62)(43,71)(44,63)(48,68)(49,79)(53,69)(54,78)(58,80)(59,77)(60,74)(61,75)(66,81)(70,72)(73,76)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6084);