Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#18208 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C107×#1595 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C2×C42×C107 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 4 generators on 139 points:
(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,20)(14,25)(17,28)(22,31)(23,29)(26,30)(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);(1,2)(3,12)(4,16)(5,10)(6,9)(7,15)(8,11)(13,20)(14,26)(17,29)(18,23)(19,28)(21,24)(22,25)(27,30)(31,32);(4,17)(7,19)(13,27)(15,28)(16,29)(20,30)(21,31)(24,32);(1,3,6,14)(2,8,9,22)(4,13,17,27)(5,11,18,25)(7,20,19,30)(10,12,23,26)(15,24,28,32)(16,21,29,31)4601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 11196);