Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12832 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #427 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C2×C109 | #1807 |
| prime radical | C2×#41×C109 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C2×M16×C109 ◁ C2×#131×C109 ◁ G | |
358 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)(3,13,16,34)(4,18,20,9)(5,22)(6,25)(7,11)(8,15,28,40)(10,23)(12,26)(14,36)(17,43,42,35)(19,47,48,51)(21,31,50,46)(24,52)(27,37)(29,56,54,41)(30,49,45,63)(32,53)(33,39)(38,57)(44,61)(55,64)(58,60)(59,62)(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);(1,3,5,15,6,17,23,41,7,16,24,40,26,42,53,56)(2,8,10,13,12,29,22,35,11,28,32,34,25,54,52,43)(4,14,19,37,21,38,49,60,20,39,48,61,50,62,63,64)(9,27,30,33,31,55,51,57,18,44,45,36,46,58,47,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,27)(10,30)(11,18)(12,31)(13,33)(15,37)(16,39)(17,38)(22,51)(23,49)(24,48)(25,46)(26,50)(28,44)(29,55)(32,45)(34,36)(35,57)(40,61)(41,60)(42,62)(43,59)(47,52)(53,63)(54,58)(56,64)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5820);