Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C107 · Schur multiplier (H2) = C27×C4 |
| direct product | C107×#34058 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | C24 |
| commutator | C22 | C24×C4×C107 |
| Frattini, cosocle | C23 | C25×C107 |
| socle | C23×C107 | C25 |
| prime radical | C23×D4×C107 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ G | |
2858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 5 generators on 127 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(5,7,8,12)(6,9,10,11);(7,12)(9,11)(14,17)(18,20);(6,10)(9,11);(13,15,16,19)(14,18,17,20)9416 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 27046);