Finite Group DatabasebetaFinite Group Databasebeta| order | 27392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#33501 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C107 | C24 |
| commutator | C22 | C23×C8×C107 |
| Frattini, cosocle | C2×C8 | C24×C107 |
| socle | C22×C107 | C23×C8 |
| prime radical | C23×C107 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C107 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 4 generators on 155 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,28)(21,29)(22,27)(24,33)(25,34)(26,35)(30,41)(31,39)(32,40)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47);(2,8)(6,9)(7,16)(14,15)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)8132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27392, 26489);