Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#12995 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1874 |
| prime radical | C17×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
476 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)(18,19,24,31,25,32,49,63,26,33,48,62,23,30,22,29)(20,34,38,56,39,60,36,53,40,67,46,27,37,54,74,55)(21,41,44,61,43,59,47,65,45,64,51,57,50,28,42,58)(35,68,71,79,70,78,69,80,72,81,76,77,75,66,73,52);(18,20,25,37,26,40,23,39)(19,27,32,53,33,56,30,55)(21,35,43,75,45,72,50,70)(22,46,24,36,49,38,48,74)(28,52,41,77,59,80,64,79)(29,60,31,34,63,54,62,67)(42,76,44,69,47,71,51,73)(57,81,58,78,61,68,65,66);(18,21)(19,28)(20,35)(22,47)(23,50)(24,51)(25,43)(26,45)(27,52)(29,61)(30,64)(31,65)(32,41)(33,59)(34,66)(36,73)(37,75)(38,76)(39,70)(40,72)(42,49)(44,48)(46,71)(53,77)(54,81)(55,79)(56,80)(57,63)(58,62)(60,68)(67,78)(69,74)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5983);