Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C2×D4 |
| commutator | C4 | C2×C4×C8×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C2×C4⋊C8 |
| prime radical | C23×C17 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,20)(22,23)(24,36,39,49)(25,41)(26,32)(27,33)(28,34)(29,35)(30,38,46,44)(31,43)(37,47,51,50)(40,45,52,53)(42,48);(22,24,27,40,28,38,43,51,29,39,42,52,26,44,25,37)(23,30,33,47,34,36,31,45,35,46,48,50,32,49,41,53);(18,20,21,19)(22,25,26,42,29,43,28,27)(23,31,32,33,35,41,34,48)(24,37,44,52,39,51,38,40)(30,45,49,47,46,53,36,50)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5644);