Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C22×C4 |
| direct product | C107×#13246 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #466 |
| commutator | C2×C8 | C22×C4×C107 |
| Frattini, cosocle | C22×C8 | C23×C107 |
| prime radical | C22×D8×C107 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ #405×C107 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 143 points:
(2,4)(5,6,11,17,12,19,31,18)(7,20,24,33,26,8,25,28)(9,29,10,15,21,16,32,36)(13,14,34,22,35,23,27,30)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,2,3,4)(5,7,21,22,11,25,32,14,12,26,9,30,31,24,10,23)(6,13,15,8,18,34,29,28,19,35,36,20,17,27,16,33);(5,8,9,27,11,28,10,13,12,20,21,34,31,33,32,35)(6,14,36,7,18,30,16,25,19,23,15,26,17,22,29,24)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 6234);