Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C24 |
| direct product | C2×C107×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C24×C107 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ C22×M16×C107 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(3,4,8,12)(5,13,14,31)(6,17,18,34)(7,22,20,11)(9,24,25,15)(10,27,28,21)(16,29,23,32)(19,26,33,30);(6,18)(10,28)(13,31)(15,24)(17,34)(19,33)(21,27)(29,32);(3,5,7,23,8,14,20,16)(4,9,11,30,12,25,22,26)(6,19,21,24,18,33,27,15)(10,29,17,31,28,32,34,13)4280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26643);