Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#12640 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D8 |
| commutator | C8 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C2×#102 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ C2×C4×C8×C17 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,20)(22,23)(24,33)(25,38)(26,41)(27,43)(28,32)(29,35)(30,40)(31,39)(34,47)(36,44)(37,53)(42,46)(45,52)(48,51)(49,50);(22,24,26,36,39,50,40,51,28,34,25,35,27,37,42,52)(23,29,30,33,43,48,41,53,32,44,46,47,31,45,38,49);(18,20,21,19)(24,34)(29,44)(33,47)(35,36)(37,50)(45,48)(49,53)(51,52)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5628);