Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#17986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22×D4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C42 | C24×C13 |
| socle | C22×C13 | C2×C4×D4 |
| prime radical | C13×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,23)(16,24,26,29)(17,22,18,21)(20,27,28,25)(30,31,34,38,35,37,43,44)(32,36,40,33,39,41,45,42);(14,16)(15,20)(17,25)(18,27)(19,26)(21,24)(22,29)(23,28)(30,32)(31,36)(33,38)(34,40)(35,39)(37,41)(42,44)(43,45);(30,33,34,41,35,42,43,36)(31,32,38,40,37,39,44,45);(14,17,19,18)(15,21,23,22)(16,25,26,27)(20,24,28,29)(31,37)(32,39)(38,44)(40,45)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 10974);