Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C17 · Schur multiplier (H2) = C2×C8 |
| direct product | C17×#8266 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C17 | D4 |
| commutator | C4 | C2×C4×C8×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C22×C17 | C4×C4⋊C4 |
| prime radical | C23×C17 | #117 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(27,31,33,36)(32,38,39,40);(18,19,20,22,21,23,24,25)(26,27)(28,32)(29,36)(30,33)(31,37)(34,40)(35,39)(38,41);(26,28,29,34,30,35,37,41)(27,32,36,40,33,39,31,38)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1254);