Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C107 · Schur multiplier (H2) = C26 |
| direct product | C2×C107×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C107 | C22⋊C4 |
| commutator | C22 | C23×C8×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C2×C22⋊C8 |
| prime radical | C22×#41×C107 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C107 ◁ C23×C8×C107 ◁ G | |
1154 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);(3,4,6,11,7,12,18,26,8,10,19,23,20,24,31,33)(5,13,15,25,16,27,29,34,14,9,17,21,28,22,30,32);(3,5)(4,9)(6,17)(7,16)(8,14)(10,13)(11,25)(12,22)(15,19)(18,30)(20,28)(21,23)(24,27)(26,34)(29,31)(32,33);(5,14)(9,13)(15,17)(16,28)(21,25)(22,27)(29,30)(32,34)9416 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26324);