Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12744 |
| 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 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2,6,13,8,12,7,29)(3,14,17,42,18,41,30,38)(4,19,21,37,22,10,28,36)(5,23,25,39,27,56,26,11)(9,24,33,48,32,57,34,16)(15,43,45,62,47,40,46,61)(20,51,53,50,54,55,58,35)(31,52,59,44,49,63,60,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,3)(2,9)(4,15)(5,24)(6,17)(7,30)(8,18)(10,31)(11,38)(12,32)(13,33)(14,23)(16,26)(19,49)(20,52)(21,45)(22,47)(25,48)(27,57)(28,46)(29,34)(35,61)(36,59)(37,60)(39,42)(40,55)(41,56)(43,51)(44,53)(50,62)(54,63)(58,64);(1,4,8,22)(2,10,12,19)(3,15,18,47)(5,20,27,54)(6,28,7,21)(9,31,32,49)(11,35,39,50)(13,37,29,36)(14,40,41,43)(16,44,48,64)(17,46,30,45)(23,55,56,51)(24,52,57,63)(25,58,26,53)(33,60,34,59)(38,61,42,62)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5732);