Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C2×#41×C17 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
386 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)(18,19,24,29)(20,30,35,51)(21,36,40,73)(22,41,44,27)(23,45,48,72)(25,34,52,32)(26,53,56,39)(28,58,61,43)(31,64,37,74)(33,50,46,59)(38,76,42,60)(47,57,77,55)(49,78,54,69)(62,75,79,66)(63,68,80,70)(65,81,67,71);(18,20)(19,25)(21,37)(22,34)(23,46)(24,35)(26,54)(27,51)(28,59)(29,52)(30,41)(31,40)(32,44)(33,48)(36,71)(38,65)(39,66)(42,67)(43,68)(45,63)(47,69)(49,56)(50,61)(53,75)(55,79)(57,62)(58,70)(60,64)(72,80)(73,81)(74,76)(77,78);(18,21,24,40)(19,26,29,56)(20,31)(22,42,44,38)(23,47,48,77)(25,49)(27,57,41,55)(28,60,61,76)(30,62)(32,67)(33,69)(34,65)(35,37)(36,72,73,45)(39,58,53,43)(46,78)(50,64)(51,79)(52,54)(59,74)(63,81)(66,70)(68,75)(71,80)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6393);