Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#33456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C22×D8 |
| commutator | C8 | C23×C4×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C22×D16×C107 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ #117×C107 ◁ #405×C107 ◁ G | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2,4,3)(5,6)(7,12)(8,11)(9,22)(10,13)(14,19)(15,20)(16,28)(17,23)(18,32)(21,24)(25,33)(26,31)(27,30)(29,35)(34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(6,11)(8,19)(9,20)(12,23)(13,24)(14,15)(17,31)(18,30)(21,33)(22,25)(26,27)(28,35)(29,36)(32,34);(1,3,4,2)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,28)(14,26)(15,27)(19,31)(20,30)(21,29)(22,32)(24,35)(25,34)(33,36);(6,13)(7,16)(11,24)(14,25)(15,22)(17,29)(18,30)(31,36)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26444);