Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C2×C107×#2074 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C107 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C107 ◁ C22×C42×C107 ◁ G | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 27392 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,30)(24,28)(25,27)(26,29);(8,11)(12,14)(15,17,18,24)(16,20,21,28)(19,23,26,25)(22,30,29,27);(4,6)(7,9,10,13)(8,12,11,14)(16,21)(17,23)(19,26)(20,27)(24,25)(28,30)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 16614);