Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16×C17 · Schur multiplier (H2) = C2×C42 |
| direct product | C17×#12590 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8×C17 | C22 |
| commutator | C2 | C2×C4×C16×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle, prime radical | C23×C17 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8×C17 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 4352 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23)(24,27)(25,26)(28,29)(30,31,32,35,33,36,38,41,34,37,39,42,40,43,44,45);(18,19,20,21)(23,26)(27,29);(22,24,25,28)(23,27,26,29)2720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5578);