Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12751 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1786 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C2×#578×C109 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G | |
876 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)(3,14,18,42)(4,20,23,38)(5,25)(6,11)(7,13)(8,16,32,48)(9,22,34,24)(10,27)(12,29)(15,43)(17,41,19,35)(21,53,56,33)(26,39,49,31)(28,36)(30,58)(37,47)(40,50)(44,59)(45,63)(46,62)(51,64)(52,55,61,57)(54,60)(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)(4,15)(5,26)(6,17)(7,18)(9,30)(10,35)(11,31)(12,39)(13,32)(14,36)(16,47)(19,29)(20,51)(21,54)(22,45)(23,46)(24,50)(25,42)(27,48)(28,49)(33,43)(34,60)(37,41)(38,59)(40,61)(44,55)(52,63)(53,62)(56,58)(57,64);(1,4,7,23)(2,9,13,34)(3,15,18,46)(5,21,28,56)(6,24,29,22)(8,30,32,60)(10,33,37,53)(11,38,12,20)(14,40,42,63)(16,44,48,64)(17,50,19,45)(25,52,36,61)(26,54,49,58)(27,57,47,55)(31,59,39,51)(35,43,41,62)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5739);