Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1804 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C23×C17 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
376 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)(20,31,35,50)(21,36)(22,27)(23,43)(24,30)(25,33,49,66)(26,38)(28,41)(29,42)(32,60,63,81)(34,48,44,55)(37,69)(39,70)(40,56)(45,51)(46,54)(47,76,79,67)(52,74)(53,73)(57,72)(58,71,77,62)(59,75,80,65)(61,78,64,68);(18,20,24,35)(19,25)(21,32,39,63)(22,33,42,66)(23,44,46,34)(26,47)(27,31)(28,55)(29,50)(30,49)(36,68)(37,61,40,64)(38,71,73,62)(41,75,72,65)(43,59)(45,76,74,67)(48,57)(51,77)(52,58)(53,79)(54,80)(56,81)(60,69)(70,78);(18,21,24,39)(19,26,30,53)(20,32,35,63)(22,40,42,37)(23,45,46,74)(25,47,49,79)(27,52,29,51)(28,56,57,69)(31,58,50,77)(33,64,66,61)(34,67,44,76)(36,54,70,43)(38,72,73,41)(48,60,55,81)(59,68,80,78)(62,75,71,65)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5815);