Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#33451 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | C2×D8 |
| commutator | C8 | C23×C4×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C22×D8 |
| prime radical | C2×#508×C107 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2,4,3)(5,6)(7,16)(8,14)(9,21)(10,25)(11,13)(12,18)(15,23)(17,33)(19,32)(20,30)(22,28)(24,35)(26,31)(27,34)(29,36)(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);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,28)(15,31)(16,21)(18,23)(20,34)(24,36)(25,29)(30,33)(32,35);(1,3,4,2)(6,13)(12,27)(14,25)(15,30)(16,32)(21,35)(28,29)(31,33);(5,8,11,10)(6,14,13,25)(7,17,19,26)(9,20,24,23)(12,28,27,29)(15,21,30,35)(16,33,32,31)(18,22,34,36)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26439);