Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#18154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | #392×C107 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C2×C42×C107 ◁ G | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 4 generators on 139 points:
(2,7)(3,11)(4,14)(8,20)(9,18)(10,15)(13,25)(16,27)(19,22)(21,30)(23,28)(29,31)(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);(1,2,5,10)(3,8,11,22)(4,15,14,7)(6,9,17,23)(12,20,24,19)(13,21,25,31)(16,28,27,18)(26,30,32,29);(4,16)(7,18)(12,26)(14,27)(15,28)(19,29)(20,30)(24,32);(1,3,6,13)(2,8,9,21)(4,12,16,26)(5,11,17,25)(7,19,18,29)(10,22,23,31)(14,24,27,32)(15,20,28,30)3959 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 11142);