Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C2×C42 |
| direct product | C13×#2825 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C13 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
522 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(19,21)(22,23)(24,27)(25,26)(28,29)(30,31,34,35)(32,36,37,33);(16,20)(19,21)(23,26)(24,28)(32,37)(33,36);(14,15)(16,19)(17,18)(20,21)(22,23)(24,29)(25,26)(27,28)(30,32)(31,33)(34,37)(35,36);(14,16)(15,19)(17,20)(18,21)(22,24)(23,27)(25,28)(26,29)(30,33)(31,32)(34,36)(35,37)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1557);