Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C22×C42 |
| direct product | C17×#22056 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C22×D4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C2×C17×#447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C2×C17×#117 ◁ G | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,20)(22,23)(24,28)(25,27)(26,29)(30,31,34,38)(32,35,39,43)(33,36,41,37)(40,42,45,44);(18,20,21,19)(22,24)(23,26)(25,29)(27,28)(30,32)(31,35)(33,40)(34,39)(36,42)(37,44)(38,43)(41,45);(24,29)(26,28)(31,36)(33,41)(35,42)(37,38)(40,45)(43,44);(23,27)(26,28)(32,39)(35,43)(40,45)(42,44)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15044);