Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C22×C4 |
| direct product | C107×#29120 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C107 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C22×#41×C107 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C107 ◁ C2×#117×C107 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,25,21,26,20,30)(19,27,28,32,29,23,31,24)(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);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,25)(26,27)(30,32);(11,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,25)(26,32)(27,30);(2,6)(5,7)(17,20,21,22)(18,25,26,30)(19,28,29,31)(23,32,27,24)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 22108);