Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#11588 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C17 |
| Frattini, cosocle | C4×C8 | C23×C17 |
| socle | C22×C17 | C2×C22⋊C8 |
| prime radical | C17×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C4×C8×C17 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,22)(21,24)(23,25)(26,27,29,34,31,35,42,49,30,33,41,46,43,47,54,56)(28,36,38,48,39,50,52,57,37,32,40,44,51,45,53,55);(19,22)(20,25)(23,24)(26,28)(27,32)(29,40)(30,37)(31,39)(33,36)(34,48)(35,45)(38,41)(42,53)(43,51)(44,46)(47,50)(49,57)(52,54)(55,56);(27,33)(28,37)(34,46)(35,47)(38,40)(39,51)(49,56)(52,53)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 4576);