Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8×C17 · Schur multiplier (H2) = C29 |
| direct product | C2×C17×#2982 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C24 |
| commutator | C2 | C24×C8×C17 |
| Frattini, cosocle | C2×C4 | C25×C17 |
| socle | C23×C17 | C23×C4 |
| prime radical | C24×C17 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ G | |
2234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(3,4,5,7,6,8,9,10)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26);(11,13,15,21)(12,16,18,19)(14,22,23,20)(17,25,26,24);(11,14,15,23)(12,17,18,26)(13,20,21,22)(16,24,19,25);(12,18)(16,19)(17,26)(24,25)2312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 53230);