Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C24×C4 |
| direct product | C17×#17573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| socle | C22×C17 | C22×C22⋊C4 |
| prime radical | C17×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C42×C17 ◁ G | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,26,28,23)(21,29)(22,25)(24,30)(27,31,33,32)(34,35)(36,44,46,49)(37,42)(38,43)(39,41)(40,45,48,47);(18,20)(19,23)(21,27)(22,28)(24,31)(25,26)(29,32)(30,33)(34,36)(35,40)(37,45)(38,47)(39,46)(41,48)(42,44)(43,49);(18,21,22,30)(19,24,25,29)(20,27,28,33)(23,31,26,32)(35,41)(40,48)(42,43)(44,49);(34,37,39,38)(35,42,41,43)(36,45,46,47)(40,44,48,49)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 10561);