Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C107 · Schur multiplier (H2) = C23×C4 |
| direct product | C107×#10800 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | C2×C42 |
| commutator | C2×C4 | C2×C42×C107 |
| Frattini, cosocle | C23×C4 | C23×C107 |
| socle | C22×C107 | #376 |
| prime radical | C24×C107 | #41 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(3,4)(5,6)(7,16,19,30)(8,13)(9,14)(10,24)(11,15)(12,18,26,20)(17,29,32,36)(21,27)(22,34)(23,28)(25,31,35,33)(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);(5,7,10,20,11,19,9,18)(6,12,14,16,15,26,24,30)(8,17,22,33,23,32,21,31)(13,25,27,29,28,35,34,36);(2,4)(5,8,9,21,11,23,10,22)(6,13,14,27,15,28,24,34)(7,17,18,31,19,32,20,33)(12,25,16,29,26,35,30,36)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 3788);