Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C27 |
| direct product | C2×C107×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C107 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C22×#41×C107 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C107 ◁ C23×C4×C107 ◁ G | |
1902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 145 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(3,4)(5,6)(7,8,13,18)(9,19,23,30)(10,24,28,33)(11,17,12,16)(14,22,31,21)(15,27,34,26)(20,35,37,29)(25,32,36,38);(4,6)(7,9)(8,14)(10,25)(11,22)(12,21)(13,23)(15,32)(16,30)(17,19)(18,31)(20,27)(24,35)(26,37)(28,36)(29,33)(34,38);(7,10,13,28)(8,15,18,34)(9,20,23,37)(11,27,12,26)(14,29,31,35)(16,33,17,24)(19,32,30,38)(21,25,22,36)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 16425);