Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#13123 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | #121 |
| commutator | C2×C4 | C22×C8×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1923 |
| prime radical | C2×#41×C17 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C22×C8×C17 ◁ G | |
312 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)(18,19,23,29,24,28,44,47,25,30,45,54,22,43,32,58)(20,31,35,56,36,50,65,75,37,57,66,78,34,26,46,49)(21,38,41,69,42,68,70,79,40,27,51,53,39,52,59,72)(33,60,63,80,64,48,73,74,62,55,76,77,61,67,71,81);(18,20,32,46,22,34,45,66,25,37,44,65,24,36,23,35)(19,26,47,56,43,57,29,49,30,50,58,78,28,31,54,75)(21,33,59,71,39,61,51,76,40,62,70,73,42,64,41,63)(27,48,72,77,68,60,53,74,38,67,79,80,52,55,69,81);(18,21,22,39,25,40,24,42)(19,27,28,52,30,38,43,68)(20,33,34,61,37,62,36,64)(23,41,32,59,45,51,44,70)(26,48,31,55,50,67,57,60)(29,53,47,72,54,69,58,79)(35,63,46,71,66,76,65,73)(49,74,56,77,75,81,78,80)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6111);