Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C22×C4 |
| direct product | C107×#32703 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C107 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| socle | C2×C107 | #2782 |
| prime radical | #575×C107 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C107 ◁ C22×C4×C107 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2,6,10,7,12,5,11)(3,13,16,8,17,24,15,25)(4,18,22,9,20,28,21,27)(14,29,30,26,31,23,19,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,24)(11,13)(12,25)(14,22)(18,29)(20,30)(21,31)(23,28)(27,32);(1,4,5,21,7,20,6,22)(2,9,10,28,12,27,11,18)(3,14,16,30,17,31,15,19)(8,23,13,26,25,29,24,32);(3,15,17,16)(4,20)(8,24,25,13)(9,27)(14,30,31,19)(18,28)(21,22)(23,26,29,32)3638 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 25691);