Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C25 |
| direct product | C2×C109×#1618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C109 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C109 |
| prime radical | C2×#538×C109 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C23×C4×C109 ◁ G | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(4,7)(5,9)(8,10)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21);(3,4)(5,10)(6,8)(7,9)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(5,9)(7,10)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 11108);