Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C17 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×C17×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C17 | D4 |
| commutator | C4 | C43×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle | C23×C17 | C2×C4⋊C4 |
| prime radical | C24×C17 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C17 ◁ C22×C42×C17 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(5,6)(7,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(6,8)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 3692);