Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C22 |
| direct product | C107×#8607 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #121 |
| commutator | C22×C4 | C22×C4×C107 |
| Frattini, cosocle | C2×C42 | C23×C107 |
| socle | C22×C107 | #380 |
| prime radical | C2×#41×C107 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C2×C42×C107 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,24,21,25,27,23)(19,26,29,22,30,31,32,28)(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);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,27,32,21,30,20,29)(18,22,24,31,25,28,23,26);(3,9)(6,12)(10,15)(13,16)(17,18)(19,28)(20,23)(21,25)(22,30)(24,27)(26,32)(29,31)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 1595);