Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C107×#33584 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C2×#509×C107 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 143 points:
(2,4)(5,6)(7,17)(8,16)(9,12)(10,23)(11,14)(13,20)(15,25)(18,34)(19,33)(21,28)(22,30)(24,32)(26,31)(27,35)(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);(1,2,3,4)(6,12)(13,27)(14,16)(15,28)(17,33)(23,30)(29,32)(31,34);(5,7)(6,13)(8,18)(9,19)(10,24)(11,26)(12,27)(14,29)(15,31)(16,32)(17,23)(20,25)(21,35)(22,36)(28,34)(30,33);(5,8,9,11)(6,14,12,16)(7,18,19,26)(10,21,22,25)(13,29,27,32)(15,30,28,23)(17,31,33,34)(20,24,35,36)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26572);