Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#9253 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #386 |
| commutator | C22×C4 | C22×C4×C107 |
| Frattini, cosocle | C2×C42 | C23×C107 |
| prime radical | C2×#459×C107 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C2×C42×C107 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2,3,7,5,8,4,6)(9,10,12,15,13,14,16,11)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,6)(3,4)(7,8)(9,11,12,14,13,15,16,10);(2,7)(3,4)(6,8)4601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 2241);