Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C17 · Schur multiplier (H2) = C29 |
| direct product | C17×#61108 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C17 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
2592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,32)(21,37)(22,28)(23,31)(24,30)(25,29)(26,35)(27,39)(33,47)(34,59)(36,55)(38,67)(40,57)(41,72)(42,56)(43,53)(44,49)(45,54)(46,51)(48,64)(50,70)(52,65)(58,62)(60,73)(61,66)(63,74)(68,78)(69,76)(71,77)(75,80)(79,81);(18,20)(19,26)(21,33)(22,41)(23,44)(24,35)(25,36)(27,47)(28,52)(29,49)(30,55)(31,32)(34,42)(37,66)(38,68)(39,62)(40,63)(43,64)(45,65)(46,61)(48,54)(50,75)(51,74)(53,59)(56,72)(57,58)(60,69)(67,81)(70,79)(71,80)(73,77)(76,78);(18,21,25,40)(19,27,29,51)(20,33,36,63)(22,38,43,71)(23,39,24,46)(26,47,49,74)(28,50,53,76)(30,37,31,57)(32,58,55,66)(34,60,65,79)(35,61,44,62)(41,68,64,80)(42,69,45,70)(48,73,72,81)(52,75,59,78)(54,77,56,67);(18,22)(19,28)(20,34)(21,38)(23,45)(24,42)(25,43)(26,48)(27,50)(29,53)(30,56)(31,54)(32,59)(33,60)(35,64)(36,65)(37,67)(39,70)(40,71)(41,44)(46,69)(47,73)(49,72)(51,76)(52,55)(57,77)(58,78)(61,80)(62,68)(63,79)(66,75)(74,81);(19,29)(26,49)(27,51)(28,53)(30,31)(32,55)(37,57)(47,74)(48,72)(50,76)(52,59)(54,56)(58,66)(67,77)(73,81)(75,78)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 54096);