Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12805 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #410 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C2×#41 | C23×C109 |
| socle | C2×C109 | #1799 |
| prime radical | #525×C109 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C2×C8×C109 ◁ C2×#131×C109 ◁ G | |
434 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)(3,11)(4,15)(5,18,19,8)(6,10)(7,12)(9,17)(13,27,28,23)(14,24)(16,22,30,29)(20,25,31,32)(21,26)(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);(1,3,4,12,6,14,17,26)(2,7,9,11,10,21,15,24)(5,13,16,25,19,28,30,32)(8,20,22,23,18,31,29,27);(2,8)(7,20)(9,22)(10,18)(11,23)(15,29)(21,31)(24,27)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5793);