Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C109 · Schur multiplier (H2) = C28 |
| direct product | C109×#20612 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | C2×#120 |
| commutator | C23 | C23×C4×C109 |
| Frattini, cosocle | C22×C4 | C24×C109 |
| socle | C2×C109 | C2×#379 |
| prime radical | C2×#585×C109 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C23×C4×C109 ◁ G | |
3640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2,5,10,6,9,18,23)(3,11,14,24,13,7,17,20)(4,8,16,22,15,21,28,31)(12,19,27,30,26,25,29,32)(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)(2,7)(4,12)(5,17)(6,13)(8,19)(9,11)(10,24)(14,18)(15,26)(16,29)(20,23)(21,25)(22,32)(27,28)(30,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,21)(10,22)(11,25)(13,26)(14,27)(17,29)(18,28)(20,30)(23,31)(24,32);(2,9)(7,11)(8,21)(10,23)(19,25)(20,24)(22,31)(30,32)6322 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 13600);