Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C25 |
| direct product | C2×C107×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | #121 |
| commutator | C2×C4 | C23×C4×C107 |
| Frattini, cosocle | C22×C4 | C24×C107 |
| socle | C22×C107 | #389 |
| prime radical | C24×C107 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C23×C4×C107 ◁ G | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 6848 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(3,4,8,12,9,13,7,23)(5,14,16,10,18,25,17,26)(6,19,21,27,22,33,20,11)(15,24,29,32,31,34,30,28);(3,5,8,16,9,18,7,17)(4,10,12,25,13,26,23,14)(6,15,21,29,22,31,20,30)(11,24,19,32,27,34,33,28);(3,6,7,20,9,22,8,21)(4,11,12,19,13,27,23,33)(5,15,17,30,18,31,16,29)(10,24,25,32,26,34,14,28)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 14171);