Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C17 · Schur multiplier (H2) = C28×C4 |
| direct product | C17×#60058 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C24 |
| commutator | C2 | C23×C42×C17 |
| Frattini, cosocle | C2×C4 | C25×C17 |
| socle | C22×C17 | C24×C4 |
| prime radical | C22×#41×C17 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ G | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23,26,30,27,31,37,43)(24,28,34,40,33,39,46,50)(25,29,36,42,35,41,47,51)(32,38,45,49,44,48,52,53);(18,19,20,21)(22,24)(23,28)(25,32)(26,34)(27,33)(29,38)(30,40)(31,39)(35,44)(36,45)(37,46)(41,48)(42,49)(43,50)(47,52)(51,53);(22,25)(23,29)(24,32)(26,36)(27,35)(28,38)(30,42)(31,41)(33,44)(34,45)(37,47)(39,48)(40,49)(43,51)(46,52)(50,53);(25,35)(29,41)(32,44)(36,47)(38,48)(42,51)(45,52)(49,53);(24,33)(28,39)(32,44)(34,46)(38,48)(40,50)(45,52)(49,53)2312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 53046);