Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C2×C107×#2211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C107 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C22×D8×C107 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 145 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(4,6)(7,8)(9,17)(10,13)(11,23)(12,26)(14,19)(15,25)(16,22)(18,33)(20,37)(21,34)(24,32)(27,31)(28,35)(29,36)(30,38);(3,4,5,6)(8,13)(14,28)(15,29)(16,26)(17,33)(23,34)(30,32)(31,37);(7,9,10,18)(8,14,13,28)(11,24,21,38)(12,27,22,20)(15,31,29,37)(16,32,26,30)(17,34,33,23)(19,36,35,25)6206 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26566);