Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#11159 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #397 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C2×C109 | #1440 |
| prime radical | C22×#41×C109 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C23×C4×C109 ◁ G | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,6,10,7,12,5,11)(3,13,15,8,16,24,21,25)(4,17,19,27,20,26,22,9)(14,23,30,28,18,31,29,32)(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)(2,8)(4,18)(5,21)(6,15)(7,16)(9,23)(10,24)(11,13)(12,25)(14,20)(17,28)(19,29)(22,30)(26,32)(27,31);(1,4,5,19,7,20,6,22)(2,9,10,26,12,27,11,17)(3,14,15,29,16,18,21,30)(8,23,13,28,25,31,24,32)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4147);