Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C17 · Schur multiplier (H2) = C27×C4 |
| direct product | C17×#60129 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C25 |
| commutator | C22 | C24×C4×C17 |
| Frattini, cosocle | C2×C4 | C25×C17 |
| socle | C22×C17 | C24×C4 |
| prime radical | C22×#41×C17 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,23)(21,22)(24,25)(26,27,30,34,31,35,41,47)(28,32,38,44,36,42,49,53)(29,33,40,46,39,45,51,55)(37,43,50,54,48,52,56,57);(19,22)(23,25)(26,28)(27,32)(29,37)(30,38)(31,36)(33,43)(34,44)(35,42)(39,48)(40,50)(41,49)(45,52)(46,54)(47,53)(51,56)(55,57);(18,20,21,24)(19,23,22,25)(28,36)(32,42)(37,48)(38,49)(43,52)(44,53)(50,56)(54,57);(26,29)(27,33)(28,37)(30,40)(31,39)(32,43)(34,46)(35,45)(36,48)(38,50)(41,51)(42,52)(44,54)(47,55)(49,56)(53,57);(29,39)(33,45)(37,48)(40,51)(43,52)(46,55)(50,56)(54,57)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 53117);