Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#12848 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C107 | C2×D4 |
| commutator | C2×C4 | C22×C8×C107 |
| Frattini, cosocle | C22×C8 | C23×C107 |
| socle | C22×C107 | C2×C22⋊C8 |
| prime radical | C24×C107 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C107 ◁ C23×C8×C107 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)8132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 5836);