Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C107 · Schur multiplier (H2) = C24×C4 |
| direct product | C107×#22267 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C107 | C24 |
| commutator | C23 | C23×C4×C107 |
| Frattini, cosocle | C22×C4 | C24×C107 |
| socle | C23×C107 | C23×C4 |
| prime radical | C107×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C107 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 135 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,20,17,21,16,25)(15,19,23,28,24,22,26,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(5,7,10,6)(8,11,9,12)(13,14,16,25,17,21,18,20)(15,22,26,28,24,19,23,27);(1,2,3,4)(5,8,10,9)(6,11,7,12)(13,15,17,24)(14,19,21,22)(16,26,18,23)(20,28,25,27);(5,9,10,8)(6,11,7,12)(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,27,22,28)6848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 15255);