Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C17 · Schur multiplier (H2) = C24×C4 |
| direct product | C17×#33324 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C24 |
| commutator | C2 | C22×C4×C8×C17 |
| Frattini, cosocle | C2×C8 | C24×C17 |
| socle | C22×C17 | C23×C8 |
| prime radical | C24×C17 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ G | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23,25,30,26,31,35,43,27,29,36,41,37,42,47,51)(24,28,33,39,34,40,46,50,32,38,44,48,45,49,52,53);(18,19,20,21)(22,24)(23,28)(25,33)(26,34)(27,32)(29,38)(30,39)(31,40)(35,46)(36,44)(37,45)(41,48)(42,49)(43,50)(47,52)(51,53);(24,32)(28,38)(33,44)(34,45)(39,48)(40,49)(46,52)(50,53);(23,29)(28,38)(30,41)(31,42)(39,48)(40,49)(43,51)(50,53)2312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26312);