Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C25 |
| direct product | C2×C107×#2219 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C107 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C107 |
| prime radical | C24×C107 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C107 ◁ C23×C4×C107 ◁ C23×C8×C107 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 145 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(3,4)(5,6)(7,8,12,16)(9,17,19,31)(10,22,24,37)(11,25,23,15)(13,26,27,20)(14,29,30,18)(21,36,35,33)(28,34,32,38);(7,9,18,34,11,21,24,26,12,19,29,38,23,35,10,20)(8,13,22,33,15,28,30,31,16,27,37,36,25,32,14,17);(4,6)(9,19)(13,27)(17,31)(20,26)(21,35)(28,32)(33,36)(34,38)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26590);