Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#24028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C107 |
| prime radical | C107×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C23×C4×C107 ◁ G | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 131 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,18)(15,19)(16,23)(20,22)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(2,5)(6,8)(9,11)(10,12)(13,20)(14,16)(15,21)(17,22)(18,23)(19,24);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,17)(13,22)(14,23)(15,24)(18,21)(19,20);(2,5)(6,8)(9,13,15,14)(10,17,19,18)(11,20,21,16)(12,22,24,23)4601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 17016);