Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C2×#99 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×Q8 | C23×C13 |
| prime radical | C2×C13×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C23×C13 ◁ C22×Q8×C13 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(14,15,18,22,19,23,34,41)(16,24,27,42,25,38,32,44)(17,28,30,40,31,21,33,39)(20,35,37,29,36,43,45,26);(14,16)(15,20)(17,29)(18,32)(19,25)(21,38)(22,37)(23,36)(24,28)(26,31)(27,34)(30,35)(33,43)(39,44)(40,42)(41,45);(16,25)(20,36)(24,38)(26,29)(27,32)(35,43)(37,45)(42,44)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 865);