Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#23366 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | C24 |
| commutator | C23 | C23×C4×C107 |
| Frattini, cosocle | C22×C4 | C24×C107 |
| socle, prime radical | C23×C107 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 27392 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,14,12,16,11,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31)(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);(3,8)(5,7)(9,11,12,13)(10,14,16,15)(17,19,22,29)(18,23,25,26)(20,28,21,27)(24,32,30,31);(1,3,4,8)(2,5,6,7)(9,11,12,13)(10,15,16,14)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(2,6)(5,7)(9,11,12,13)(10,15,16,14)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 16354);