Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C82×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#7138 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C17 | D4 |
| commutator | C4 | C82×C17 |
| Frattini, cosocle | C2×C4×C8 | C22×C17 |
| socle | C22×C17 | #336 |
| prime radical | C23×C17 | #97 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
278 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(26,27,29,34,30,33,37,40,31,35,38,41,28,32,36,39);(18,19,20,22,21,23,24,25)(27,32,35,33)(34,39,41,40)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 126);