Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#9048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #120 |
| commutator | C22×C4 | C22×C4×C107 |
| Frattini, cosocle | C2×C42 | C23×C107 |
| socle | C22×C107 | #386 |
| prime radical | C23×C107 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C2×C42×C107 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,5,7,6,10,4,9)(3,11,13,8,14,15,12,16)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31)(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);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 2036);