Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C26 |
| direct product | C17×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C2×C17×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 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)(21,35,38,26)(22,27)(23,42)(24,29)(25,33)(28,45)(31,58,61,56)(32,57)(34,54)(36,65,69,49)(37,50,72,66)(39,73)(40,55)(41,53)(43,48)(44,52)(46,60,64,70)(47,63)(51,62)(59,75,79,78)(67,74,80,81)(68,76,71,77);(18,20,24,34)(19,25,29,48)(21,31,38,61)(22,39,41,63)(23,43,45,33)(26,46,35,64)(27,51,53,57)(28,54,42,30)(32,44,62,40)(36,67,69,80)(37,70,72,60)(47,55,73,52)(49,75,65,78)(50,58,66,56)(59,71,79,68)(74,77,81,76);(18,21)(19,26)(20,31)(22,36)(23,37)(24,38)(25,46)(27,49)(28,50)(29,35)(30,56)(32,59)(33,60)(34,61)(39,67)(40,68)(41,69)(42,66)(43,70)(44,71)(45,72)(47,74)(48,64)(51,75)(52,76)(53,65)(54,58)(55,77)(57,78)(62,79)(63,80)(73,81);(18,22,24,41)(19,27,29,53)(20,32,34,62)(21,36,38,69)(23,44,45,40)(25,47,48,73)(26,49,35,65)(28,55,42,52)(30,57,54,51)(31,59,61,79)(33,63,43,39)(37,71,72,68)(46,74,64,81)(50,77,66,76)(56,78,58,75)(60,80,70,67)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15445);