Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C107 · Schur multiplier (H2) = C2 |
| direct product | C107×#7524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C107 | #1404 |
| commutator | #125 | C2×C4×C107 |
| Frattini, cosocle | #490 | C22×C107 |
| prime radical | #127×C107 | D4 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C107 ◁ C22×C107 ◁ C2×C4×C107 ◁ C42×C107 ◁ #490×C107 ◁ G | |
134 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 2 generators on 139 points:
(1,2)(3,11)(4,15,17,8)(5,10)(6,9,19,27)(7,12,22,30)(13,29)(14,28,20,32)(16,23,31,24)(18,25,21,26)(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);(1,3,5,13)(2,7,10,24)(4,16,18,12)(6,20,19,14)(8,11,26,29)(9,23,27,22)(15,32,25,28)(17,30,21,31)2033 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 512);