Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1790 |
| commutator | C2×Q8 | C22×C4×C17 |
| Frattini, cosocle | C22×Q8 | C23×C17 |
| prime radical | C17×#3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C23×C17 ◁ C22×Q8×C17 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,23,26,24,29,22,28)(20,30,32,48,33,44,38,47)(21,27,36,45,37,46,35,43)(25,39,41,31,42,49,40,34);(18,20)(19,25)(21,34)(22,38)(23,32)(24,33)(26,40)(27,44)(28,41)(29,42)(30,46)(31,37)(35,49)(36,39)(43,47)(45,48);(19,26)(22,23)(25,40)(27,45)(28,29)(30,47)(32,38)(35,36)(39,49)(41,42)(43,46)(44,48)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5769);