Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#11314 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C107 | #388 |
| commutator | C22×C4 | C22×C4×C107 |
| Frattini, cosocle | C23×C4 | C23×C107 |
| socle | C2×C107 | #1443 |
| prime radical | C24×C107 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C107 ◁ C23×C4×C107 ◁ G | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 3424 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,6,11,7,10,5,12)(3,13,15,24,17,8,16,25)(4,18,21,27,22,28,20,9)(14,26,19,29,30,32,31,23)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3,5,15,7,17,6,16)(2,8,12,25,10,13,11,24)(4,19,20,30,22,31,21,14)(9,26,28,29,27,32,18,23);(1,4)(2,9,10,27)(3,14)(5,20)(6,21)(7,22)(8,23,13,29)(11,18,12,28)(15,19)(16,31)(17,30)(24,32,25,26)4922 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 4302);