Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12745 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #418 |
| 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 | |
890 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)(4,18,20,9)(5,22)(6,28)(7,11)(8,15)(10,26)(12,25)(14,39,42,60)(16,40)(17,46)(19,49,24,57)(21,34,51,48)(23,38)(27,54)(29,35)(30,41,47,44)(31,56)(32,45)(33,52,36,58)(37,59)(43,62,53,64)(50,61,55,63)(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)(2,8)(4,14)(5,23)(6,29)(7,16)(9,30)(10,35)(11,32)(12,38)(13,22)(15,26)(17,25)(18,47)(19,50)(20,42)(21,53)(24,55)(27,56)(28,31)(33,62)(34,63)(36,64)(37,46)(39,49)(40,54)(41,52)(43,51)(44,58)(45,59)(48,61)(57,60);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,30)(10,36)(11,18)(12,34)(13,39)(15,44)(16,42)(17,43)(19,27)(22,49)(23,55)(25,51)(26,58)(28,48)(29,53)(31,61)(32,47)(33,37)(35,64)(38,63)(40,60)(41,45)(46,62)(50,56)(52,59)(54,57)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5733);