Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#9372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #386 |
| commutator | C42 | C22×C4×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| prime radical | #532×C109 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 157 points:
(1,2,7,10,5,4,6,9)(3,11,13,15,14,8,12,16)(17,18,22,26,23,27,21,37)(19,28,32,39,31,43,30,24)(20,33,35,41,36,25,34,40)(29,42,46,38,45,48,44,47)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,3,6,13,5,14,7,12)(2,8,10,15,4,11,9,16)(17,19)(18,24,27,39)(20,29)(21,30)(22,32)(23,31)(25,38,33,47)(26,28,37,43)(34,44)(35,46)(36,45)(40,48,41,42);(1,4,5,2)(3,11,14,8)(6,9,7,10)(12,15,13,16)(17,20,23,36)(18,25,27,33)(19,29,31,45)(21,35,22,34)(24,38,39,47)(26,41,37,40)(28,42,43,48)(30,46,32,44)4033 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 2360);