Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#22384 |
| 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 | |
578 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)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(2,5)(3,8)(17,19,21,27,22,18,23,25)(20,24,29,31,28,26,30,32);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(18,19)(20,28)(25,27)(29,30)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 15372);