Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#33434 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C107 | C23 |
| commutator | C22 | C23×C8×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C23×C8 |
| prime radical | #532×C107 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C107 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2)(3,4)(5,6,10,15,11,16,25,32,9,14,23,30,24,31,27,36)(7,17,19,33,20,21,29,35,18,12,26,28,34,8,13,22)(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);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,20)(13,27)(14,17)(15,28)(21,31)(22,32)(23,26)(24,34)(25,29)(30,33)(35,36);(5,8,10,22,11,17,25,33,9,21,23,35,24,12,27,28)(6,13,15,7,16,19,32,20,14,29,30,18,31,26,36,34);(6,14)(7,18)(13,29)(15,30)(16,31)(19,26)(20,34)(32,36)11984 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26422);