Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#13124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C2×D4 |
| commutator | C2×C4 | C22×C8×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C8×D4 |
| prime radical | C2×#41×C17 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C22×C16×C17 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,28,34,27,32,25,43,29,35,44,31,26,33,37,50)(24,36,41,52,40,49,38,46,42,47,53,51,39,30,45,48);(22,24,37,45,26,39,44,53,29,42,25,38,27,40,28,41)(23,30,43,52,33,47,34,48,35,49,50,51,32,36,31,46);(19,21)(22,25,26,28,29,37,27,44)(23,31,32,50,35,34,33,43)(24,38,39,41,42,45,40,53)(30,46,36,51,49,48,47,52)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6112);