Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C107 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#504×C107 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16×C107 | C22 |
| commutator | C2 | C22×C4×C8×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C23×C8 |
| prime radical | C2×#41×C107 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16×C107 ◁ G | |
834 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)(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);(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(5,7)(6,11)(8,16)(9,17)(10,15)(12,22)(13,23)(14,21)(18,29)(19,27)(20,28)(24,33)(25,31)(26,32)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36)17120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26313);