Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#22576 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C22×D4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C22×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C4×C22⋊C4×C17 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2176 |
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)(18,19,20,21)(22,23)(24,26)(25,27)(28,29)(31,36)(33,34)(37,43)(38,42)(39,40)(44,45);(22,24,25,28)(23,26,27,29)(30,31,35,38)(32,37,41,44)(33,36,34,42)(39,43,40,45);(23,27)(26,29)(30,32,35,41)(31,37,38,44)(33,39,34,40)(36,43,42,45);(30,33)(32,39)(34,35)(36,42)(40,41)(43,45)1190 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15564);