Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C27 |
| direct product | C13×#23412 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| socle | C2×C13 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C23×C4×C13 ◁ G | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
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)(16,26)(17,31)(18,23)(19,24)(20,25)(21,29)(22,32)(27,35)(28,36)(30,40)(33,39)(34,38)(37,43)(41,42)(44,45);(14,16)(15,21)(17,27)(18,28)(19,29)(20,30)(22,37)(23,26)(24,40)(25,36)(31,41)(32,42)(33,44)(34,43)(35,39)(38,45);(14,17)(15,22)(16,27)(18,32)(19,33)(20,34)(21,35)(23,38)(24,31)(25,39)(26,41)(28,42)(29,44)(30,43)(36,37)(40,45);(14,18,20,19)(15,23,25,24)(16,28,30,29)(17,32,34,33)(21,26,36,40)(22,38,39,31)(27,42,43,44)(35,41,37,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 16400);