Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C71 · Schur multiplier (H2) = C22×C4 |
| direct product | C71×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | #442 |
| commutator | C8 | C2×C42×C71 |
| Frattini, cosocle | C22×C8 | C23×C71 |
| prime radical | C23×C71 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#97×C71 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 3 generators on 107 points:
(1,2)(3,4)(5,6,9,14)(7,17)(8,23,10,26)(11,28,24,16)(12,20)(13,27,15,25)(18,33)(19,32)(21,36)(22,30)(29,34)(31,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(2,4)(5,7,10,20,11,21,27,34,9,22,8,19,24,18,25,35)(6,12,15,30,16,31,26,36,14,32,13,17,28,29,23,33);(7,18,22,21)(8,10)(12,29,32,31)(13,15)(17,33,30,36)(19,35,20,34)(23,26)(25,27)3266 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 5636);