Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#29085 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C109 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C109 |
| prime radical | C22×D4×C109 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C4×C4⋊C4×C109 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,23,29,22)(20,26,27,24)(28,31,30,32)(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);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,27,28,21,29,20,30)(18,22,24,32,25,23,26,31);(10,14)(13,15)(17,19,21,29)(18,23,25,22)(20,28,27,30)(24,32,26,31);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20,21,27)(18,24,25,26)(19,28,29,30)(22,31,23,32)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 22073);