Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C107×#24288 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C107 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| socle | C2×C107 | #2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C107 ◁ C22×C4×C107 ◁ G | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2)(3,15)(4,20)(5,23)(6,13)(7,12)(8,14)(9,18)(10,22)(11,24)(16,27)(17,32)(19,31)(21,28)(25,30)(26,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3,6,18,8,19,7,21)(2,9,12,15,14,28,13,31)(4,16,22,5,17,24,26,25)(10,11,20,23,29,27,32,30);(1,4)(2,10)(3,16)(5,18)(6,22)(7,26)(8,17)(9,27)(11,28)(12,20)(13,32)(14,29)(15,30)(19,24)(21,25)(23,31);(1,5,6,24,8,25,7,16)(2,11,12,23,14,27,13,30)(3,17,18,26,19,4,21,22)(9,20,15,29,28,32,31,10)4280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 17276);