Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#21995 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C22×D4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| socle | C22×C17 | C22×C4⋊C4 |
| prime radical | C24×C17 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C17×#117 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,22)(20,24)(26,27,31,35,32,36,30,45)(28,33,40,48,38,47,39,54)(29,41,43,49,44,55,42,34)(37,50,53,56,51,57,52,46);(18,19)(20,23)(21,22)(24,25)(26,28)(27,33)(29,37)(30,39)(31,40)(32,38)(34,46)(35,48)(36,47)(41,50)(42,52)(43,53)(44,51)(45,54)(49,56)(55,57);(19,22)(23,25)(26,29,30,42,32,44,31,43)(27,34,35,41,36,49,45,55)(28,37,39,52,38,51,40,53)(33,46,48,50,47,56,54,57);(18,20,21,24)(19,23,22,25)(28,38)(33,47)(37,51)(39,40)(46,56)(48,54)(50,57)(52,53)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 14983);