Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#33734 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C22×D8 |
| prime radical | C23×C107 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 147 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,23,35)(12,24,26,39)(13,27,25,17)(15,28,30,21)(16,31,32,20)(22,38,37,34)(29,36,33,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(2,5)(6,8);(9,11,20,36,13,22,26,28,14,23,31,40,25,37,12,21)(10,15,24,34,17,29,32,35,18,30,39,38,27,33,16,19);(1,3,4,7)(2,6,5,8)8132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26722);