Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C2×C107×#2194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C2×#576×C107 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ C2×#131×C107 ◁ G | |
1130 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);(3,4,9,15)(5,16,19,32)(6,20,24,31)(7,14,8,13)(10,26,29,17)(11,25,30,18)(12,22,27,23)(21,33,34,28);(3,5)(4,10)(6,21)(7,18)(8,25)(9,19)(11,27)(12,30)(13,28)(14,33)(15,29)(16,20)(17,23)(22,26)(24,34)(31,32);(3,5)(4,11)(6,17)(7,25)(8,18)(9,19)(10,27)(12,29)(13,32)(14,16)(15,30)(20,33)(21,23)(22,34)(24,26)(28,31)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26523);