Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C107×#24462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C2×#578×C107 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ G | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2,3,4)(5,6,11,15)(7,17,20,27)(8,21,25,13)(9,19,12,28)(10,16,29,14)(18,34,36,31)(22,30,35,23)(24,32,26,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(5,7,11,20)(6,12,15,9)(8,22,25,35)(10,28,29,19)(13,31,21,34)(14,27,16,17)(18,24,36,26)(23,32,30,33);(2,4)(5,8,11,25)(6,13,15,21)(7,18,20,36)(9,23,12,30)(10,24,29,26)(14,32,16,33)(17,31,27,34)(19,35,28,22);(5,9)(6,7)(8,23)(10,17)(11,12)(13,18)(14,28)(15,20)(16,19)(21,36)(22,32)(24,31)(25,30)(26,34)(27,29)(33,35)4601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 17450);