Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C22 |
| direct product | C73×#12897 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #117 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | #432 |
| prime radical | C23×C73 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×#97×C73 ◁ G | |
278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,13,19,29,36,15,21,31,17,30,18,12,27)(11,22,25,37,24,16,23,32,26,33,38,40,39,34,28,35)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3,4,7)(2,5,6,8)(9,11,15,26)(10,16)(12,28,29,23)(13,24,30,39)(14,25,31,38)(17,35)(18,22)(19,33)(20,32)(21,34)(27,37)(36,40);(2,6)(5,8)(9,12,13,14,15,29,30,31)(10,17,18,36,21,20,19,27)(11,23,24,38,26,28,39,25)(16,32,22,37,34,35,33,40)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5885);