Finite Group DatabasebetaFinite Group Databasebeta| order | 26973 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C37 · Schur multiplier (H2) = C3 |
| direct product | C37×#10624503 |
| subgroup | G/H | |
|---|---|---|
| center | C27×C37 | H(3,𝔽3) |
| commutator | C32 | C3×C27×C37 |
| Frattini, cosocle | C3×C27 | C32×C37 |
| socle | C3×C37 | C32⋊C27 |
| prime radical | H(3,𝔽3)×C37 | C27 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C37 ◁ C3×C27×C37 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 280 |
| transitive degree | μt(G) = 8991 |
A minimal faithful permutation representation — 2 generators on 280 points:
(1,2,7,4,9,25,18,33,64,5,10,26,19,34,65,54,81,122,21,36,67,56,83,124,111,144,181,6,11,27,20,35,66,55,82,123,22,37,68,57,84,125,112,145,182,59,86,127,114,147,184,173,200,222,23,38,69,58,85,126,113,146,183,60,87,128,115,148,185,174,201,223,116,149,186,175,202,224,218,232,240)(3,12,39,15,41,89,49,93,152,16,42,90,50,94,153,106,158,206,52,95,154,108,159,207,170,210,235,17,8,28,51,30,71,107,76,131,53,31,72,109,77,132,171,139,190,110,79,133,172,141,191,217,197,227,14,32,24,47,78,62,103,140,119,48,80,63,104,142,120,168,198,179,105,143,121,169,199,180,216,231,221)(13,29,61,44,74,117,98,136,176,45,75,118,99,137,177,163,195,219,100,138,178,164,196,220,213,230,239,43,73,88,96,134,150,160,192,203,97,135,151,161,193,204,211,228,233,162,194,205,212,229,234,237,242,243,46,40,70,101,91,129,165,155,187,102,92,130,166,156,188,214,208,225,167,157,189,215,209,226,238,236,241)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280);(1,3,13)(2,8,29)(4,15,44)(5,16,45)(6,17,43)(7,24,61)(9,30,74)(10,31,75)(11,32,73)(12,40,38)(14,46,23)(18,49,98)(19,50,99)(20,51,96)(21,52,100)(22,53,97)(25,62,117)(26,63,118)(27,39,88)(28,70,69)(33,76,136)(34,77,137)(35,78,134)(36,79,138)(37,80,135)(41,91,85)(42,92,87)(47,101,58)(48,102,60)(54,106,163)(55,107,160)(56,108,164)(57,109,161)(59,110,162)(64,119,176)(65,120,177)(66,89,150)(67,121,178)(68,90,151)(71,129,126)(72,130,128)(81,139,195)(82,140,192)(83,141,196)(84,142,193)(86,143,194)(93,155,146)(94,156,148)(95,157,149)(103,165,113)(104,166,115)(105,167,116)(111,170,213)(112,171,211)(114,172,212)(122,179,219)(123,152,203)(124,180,220)(125,153,204)(127,154,205)(131,187,183)(132,188,185)(133,189,186)(144,197,230)(145,198,228)(147,199,229)(158,208,201)(159,209,202)(168,214,174)(169,215,175)(173,217,237)(181,221,239)(182,206,233)(184,207,234)(190,225,223)(191,226,224)(200,231,242)(210,236,232)(216,238,218)(222,235,243)(227,241,240)5661 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26973, 63);