Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#33505 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C107 | C24 |
| commutator | C22 | C23×C8×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C23×C8 |
| prime radical | C2×#41×C107 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C107 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 131 points:
(2,5)(3,7)(9,10,13,17,14,16,21,23,12,15,20,22,11,18,19,24)(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);(1,2)(3,6)(4,5)(7,8)(10,15)(16,18)(17,22)(23,24);(2,5)(6,8)(10,15)(16,18)(17,22)(23,24);(1,3,4,7)(2,6,5,8)(9,11,12,14)(10,16,15,18)(13,19,20,21)(17,23,22,24)10700 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26493);