Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C23×C4 |
| direct product | C107×#29140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C107 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C22×D4×C107 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C107 ◁ C43×C107 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,24,21,25,28,23)(19,22,26,30,27,31,32,29)(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);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(11,15)(13,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,5,3)(2,7,8,6)6206 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 22128);