Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C107 · Schur multiplier (H2) = C22×C4 |
| direct product | C107×#7975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#95 |
| commutator | C2×C4 | C2×C42×C107 |
| Frattini, cosocle | C2×C42 | C23×C107 |
| prime radical | C24×C107 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C42×C107 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 3 generators on 139 points:
(2,7,9,20)(3,11)(4,15)(8,18,19,30)(10,24,23,31)(13,27)(14,26)(16,17)(21,22,32,28)(25,29)(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);(1,2,4,10,6,9,17,23)(3,7,13,21,14,20,27,32)(5,8,16,22,12,19,15,28)(11,18,25,31,26,30,29,24);(2,8)(7,18)(9,19)(10,22)(20,30)(21,31)(23,28)(24,32)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 963);