Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C73 ⋊ #20579 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×#64408 |
| commutator, socle | C23×C73 | C23×C4 |
| Fitting, prime radical | C26×C73 | C4 |
| cosocle | C22×Dic73 | C24 |
| hypercenter | C26 | #64408 |
| hypocenter | C73 | #20579 |
| LCS | G ▷ C23×C73 ▷ C73 | |
| UCS | C1 ◁ C23 ◁ C26 | |
5990 conjugacy classes of subgroups; only the 504 normal subgroups are shown.
This group has 5990 conjugacy classes of subgroups — too many to draw. Only its 504 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 89 points:
(10,13)(14,16)(17,18,21,26,35,48,65,50,33,45,60,76,85,89,88,83,71,55,42,31,40,53,70,78,58,41,28,19,23,30,39,52,69,82,67,46,62,79,61,77,86,81,68,51,38,29,22,27,36,49,66,74,54,37,24,32,43,56,72,84,87,80,63,75,59,44,57,73,64,47,34,25,20);(6,8);(2,4);(1,2)(3,4)(5,6)(7,8)(9,10,11,14,12,13,15,16)(18,19,20,22)(21,24,25,31)(23,28,29,27)(26,33,34,44)(30,37,38,40)(32,41,42,36)(35,46,47,61)(39,50,51,57)(43,54,55,53)(45,58,59,49)(48,63,64,76)(52,67,68,77)(56,65,71,73)(60,74,75,70)(62,78,79,66)(69,80,81,85)(72,82,83,86)(84,87,88,89)Polycyclic presentation:
2^8,73|a^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,i^a=i^27,i^g=i^721240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1104388);