Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1790 |
| commutator | C2×D4 | C22×C4×C109 |
| Frattini, cosocle | C2×#41 | C23×C109 |
| prime radical | C109×#3551 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,5,10,6,8,18,23)(3,11,14,27,13,28,31,32)(4,15,17,26,16,9,19,24)(7,20,22,12,21,29,25,30)(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);(1,3,6,13)(2,7,8,21)(4,12,16,30)(5,14,18,31)(9,11,15,28)(10,25,23,22)(17,29,19,20)(24,32,26,27);(2,8)(5,18)(7,21)(14,31)(17,19)(20,29)(24,26)(27,32)3706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5758);