Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1786 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C2×C13×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C4×C13 ◁ C2×C13×#131 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,25,27,21)(17,20)(18,32)(19,23)(22,29)(24,30)(26,38,43,33)(28,37,42,40)(31,34)(35,44)(36,41,39,45);(15,20)(17,29)(18,30)(21,33)(22,24)(23,34)(25,38)(26,41)(28,42)(31,44)(32,35)(36,37)(39,40)(43,45);(14,16)(15,21)(17,26)(18,28)(19,27)(20,33)(22,36)(23,25)(24,37)(29,41)(30,42)(31,43)(32,40)(34,38)(35,39)(44,45)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5729);