Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C2×D8 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C22×D8×C17 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C2×C42×C17 ◁ C2×C4×C8×C17 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,21)(22,23,24,25)(26,27)(28,36)(29,41)(30,33)(31,44)(32,38)(34,37)(35,42)(39,48)(40,55)(43,56)(45,54)(46,57)(47,52)(49,51)(50,53);(23,25)(26,28,37,51,31,40,43,54,30,39,52,57,42,53,29,38)(27,32,41,50,35,46,47,48,33,45,56,55,44,49,34,36);(18,19,20,21)(27,33)(32,45)(34,47)(35,44)(36,48)(41,56)(46,49)(50,55)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6192);