Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C109 · Schur multiplier (H2) = C2×C8 |
| direct product | C109×#8158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| socle | C22×C109 | C4×C22⋊C4 |
| prime radical | C24×C109 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C2×C4×C8×C109 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 157 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,25,22,26,33,39)(19,27,30,41,29,23,34,36)(20,24,31,37,32,38,44,47)(28,40,43,48,42,35,45,46)(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);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,35,38,40)(25,36,39,41)(31,43)(32,42)(33,34)(37,46,47,48)(44,45);(2,7)(6,12)(8,14)(13,16)(17,20,21,31,22,32,33,44)(18,24,25,37,26,38,39,47)(19,28,30,43,29,42,34,45)(23,35,36,46,27,40,41,48)6976 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 1146);