Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C107 · Schur multiplier (H2) = C23×C4 |
| direct product | C107×#10872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #121 |
| commutator | C2×C4 | C2×C42×C107 |
| Frattini, cosocle | C23×C4 | C23×C107 |
| socle | C22×C107 | #380 |
| prime radical | C24×C107 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C23×C4×C107 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 143 points:
(5,6,10,14,11,15,9,25)(7,16,18,12,20,27,19,28)(8,21,23,29,24,35,22,13)(17,26,31,34,33,36,32,30)(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);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,20)(13,26)(14,27)(15,28)(16,25)(21,34)(22,32)(23,31)(24,33)(29,36)(30,35);(5,8,9,22,11,24,10,23)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 3860);