Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C17 | D4 |
| commutator | C4 | C22×C16×C17 |
| Frattini, cosocle | C4×C8 | C23×C17 |
| socle | C22×C17 | C2×C4⋊C8 |
| prime radical | C23×C17 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,27)(21,25)(22,31)(23,26)(24,29)(28,32)(30,33)(34,35,36,39,37,40,42,45,38,41,43,46,44,47,48,49);(18,20,22,29,23,30,21,28)(19,24,25,27,26,32,31,33);(18,21,23,22)(19,25,26,31)(20,28,30,29)(24,27,32,33)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5000);