Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C107 · Schur multiplier (H2) = C27 |
| direct product | C107×#34974 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C107 |
| socle | C23×C107 | C25 |
| prime radical | C2×#121×C107 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ G | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 5 generators on 131 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,12,16)(17,18,20,21)(19,22,24,23)(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);(2,6)(7,8)(10,14)(15,16)(17,18)(19,23)(20,21)(22,24);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(18,21)(22,23);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,19,20,24)(18,22,21,23);(9,12,13,11)(10,16,14,15)8132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 27962);