Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C107 · Schur multiplier (H2) = C2×C4 |
| direct product | C107×#11861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | C2×#102 |
| commutator | C2×C8 | C22×C4×C107 |
| Frattini, cosocle | C22×C8 | C23×C107 |
| prime radical | #466×C107 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C2×C42×C107 ◁ C2×C4×C8×C107 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 155 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,22,28,24,29,23,39)(19,30,33,47,34,45,40,41)(20,27,21,37,36,46,35,26)(25,38,42,48,44,31,43,32)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,25)(20,32)(21,38)(22,33)(23,40)(24,34)(26,45)(27,41)(28,42)(29,44)(30,37)(31,35)(36,48)(39,43)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,35,24,36,22,21)(18,26,28,27,29,37,39,46)(19,31,33,32,34,38,40,48)(25,41,43,45,44,47,42,30)5564 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 4849);