Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C107 · Schur multiplier (H2) = C22 |
| direct product | C107×#13645 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C107 | C23 |
| commutator | C22 | C22×C16×C107 |
| Frattini, cosocle | C2×C16 | C23×C107 |
| socle | C22×C107 | C22×C16 |
| prime radical | C2×D4×C107 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C107 ◁ G | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(3,4)(5,6,8,13,9,14,20,28,10,7,16,17,22,18,31,32,11,15,21,29,23,30,34,36,24,19,12,25,26,33,27,35)(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);(5,7,8,17,9,18,20,32,10,15,16,29,22,30,31,36,11,19,21,25,23,33,34,35,24,6,12,13,26,14,27,28)11984 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 6633);