Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C107×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#99 |
| commutator | C2×C4 | C23×C4×C107 |
| Frattini, cosocle | C2×D4 | C24×C107 |
| prime radical | C22×#41×C107 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ C22×#41×C107 ◁ G | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(3,4)(5,13,16,28)(6,10)(7,20,22,11)(8,12)(9,15)(14,29,17,31)(18,33,34,26)(19,27)(21,24)(23,30)(25,32);(3,5,8,16)(4,9,12,24)(6,17,19,14)(7,21,22,15)(10,25,27,23)(11,28,20,13)(18,30,34,32)(26,29,33,31);(3,6,8,19)(4,10,12,27)(5,14,16,17)(7,18,22,34)(9,23,24,25)(11,26,20,33)(13,29,28,31)(15,30,21,32)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26345);