Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#9011 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #120 |
| commutator | C22×C4 | C22×C4×C107 |
| Frattini, cosocle | C23×C4 | C23×C107 |
| socle | C22×C107 | #386 |
| prime radical | C22×D4×C107 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C23×C4×C107 ◁ G | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 131 points:
(2,5,6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22)(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);(1,2,3,7,4,6,8,5)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 1999);