Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#19401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C109 |
| prime radical | C109×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2,6,9)(3,10,13,16)(4,8,5,14)(7,11,15,12)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,23)(22,30)(24,31)(29,32)5341 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 12389);