Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C109 · Schur multiplier (H2) = C24×C4 |
| direct product | C109×#22513 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C109 |
| Frattini, cosocle | C24 | C24×C109 |
| prime radical | C22×D4×C109 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4⋊C4×C109 ◁ G | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 4 generators on 133 points:
(2,5)(6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,2,4,5)(3,7,8,6)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,3,4,8)(2,6,5,7)5995 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 15501);