Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#12597 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | #117 |
| commutator | C2×C4 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1746 |
| prime radical | C23×C17 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C17×#405 ◁ G | |
258 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)(18,19,22,27,23,29,39,51,24,28,40,50,41,52,66,76)(20,30,33,54,34,55,59,78,32,25,42,44,58,45,68,71)(21,26,36,47,38,48,64,74,37,49,63,75,65,69,80,81)(31,53,57,77,35,60,62,79,56,43,67,70,61,46,72,73);(18,20)(19,25,28,30)(21,35,37,61)(22,33)(23,34)(24,32)(26,46)(27,44,50,54)(29,45,52,55)(31,38,56,65)(36,62,63,72)(39,59)(40,42)(41,58)(43,69)(47,73)(48,53)(49,60)(51,71,76,78)(57,64,67,80)(66,68)(70,81)(74,77)(75,79);(18,21)(19,26)(20,31,32,56)(22,36)(23,38)(24,37)(25,43,30,53)(27,47)(28,49)(29,48)(33,57,42,67)(34,35,58,61)(39,64)(40,63)(41,65)(44,70,54,77)(45,46,55,60)(50,75)(51,74)(52,69)(59,62,68,72)(66,80)(71,73,78,79)(76,81)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5585);