Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#13308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C107 |
| prime radical | C107×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ C2×#131×C107 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 143 points:
(5,6,11,17)(7,18,21,34)(8,22,26,33)(9,16,10,15)(12,28,31,19)(13,27,32,20)(14,24,29,25)(23,35,36,30)(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);(1,2)(3,4)(5,7)(6,12)(8,23)(9,20)(10,27)(11,21)(13,29)(14,32)(15,30)(16,35)(17,31)(18,22)(19,25)(24,28)(26,36)(33,34);(2,4)(5,7)(6,13)(8,19)(9,27)(10,20)(11,21)(12,29)(14,31)(15,34)(16,18)(17,32)(22,35)(23,25)(24,36)(26,28)(30,33)4280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 6296);