Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12747 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #447 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C2×C109 | #1786 |
| prime radical | C109×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2)(3,14)(4,19,22,37)(5,23)(6,12)(7,29)(8,13)(9,16)(10,21,36,28)(11,26)(15,43,46,63)(17,41)(18,42)(20,35,53,61)(24,32)(25,39)(27,55)(30,38)(31,58,59,51)(33,56)(34,48)(40,45,62,47)(44,49,64,60)(50,57,54,52)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,3)(2,9)(4,15)(5,24)(6,17)(7,30)(8,18)(10,31)(11,38)(12,32)(13,34)(14,23)(16,26)(19,49)(20,51)(21,45)(22,46)(25,48)(27,56)(28,47)(29,33)(35,43)(36,59)(37,60)(39,41)(40,54)(42,55)(44,52)(50,62)(53,58)(57,64)(61,63);(1,4)(2,10)(3,15)(5,20)(6,28)(7,21)(8,22)(9,31)(11,35)(12,37)(13,36)(14,40)(16,44)(17,47)(18,46)(19,29)(23,54)(24,51)(25,57)(26,52)(27,53)(30,45)(32,60)(33,49)(34,59)(38,43)(39,61)(41,63)(42,62)(48,64)(50,55)(56,58)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5735);