Finite Group DatabasebetaFinite Group Databasebeta| order | 19683 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C93 · Schur multiplier (H2) = C35×C92 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32×C92 |
| commutator | C9 | C3×C93 |
| Frattini, cosocle | C33×C9 | C34 |
| socle | C32 | C32×#6855 |
| prime radical | C35 | C3×H(3,𝔽3) |
| LCS | G ▷ C9 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C33×C9 ▷ C3 ▷ C1 | |
22204 conjugacy classes of subgroups; only the 8890 normal subgroups are shown.
This group has 22204 conjugacy classes of subgroups — too many to draw. Only its 8890 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 252 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 252 points:
(10,11,16)(12,17,34)(13,26,63,28,65,129,30,66,130)(14,20,37)(15,19,38)(18,44,73,46,100,142,47,101,143)(21,39,77)(22,54,114,56,116,190,58,117,191)(23,41,79)(24,43,80)(25,42,81)(27,68,136,69,137,99,71,140,173)(29,67,83,72,134,84,74,135,35)(31,50,86)(32,49,87)(33,48,36)(40,91,124,93,165,203,94,166,204)(45,102,141,103,174,70,104,175,138)(51,107,177,110,180,229,109,179,228)(52,89,146)(53,90,148)(55,119,197,121,199,164,120,198,224)(57,118,150,123,195,151,125,196,78)(59,95,147)(60,96,152)(61,98,153)(62,97,154)(64,131,206,132,82,139,133,155,207)(75,106,157)(76,105,85)(88,158,188,161,219,242,160,218,239)(92,167,200,169,226,122,168,225,201)(108,182,235,183,236,217,186,240,248)(111,181,210,187,234,209,185,232,145)(112,162,211)(113,163,213)(115,192,243,194,149,202,193,214,244)(126,170,212)(127,171,215)(128,172,216)(144,176,156)(159,220,241,221,249,184,222,250,237)(178,230,251,231,208,238,233,245,252)(189,223,246)(205,227,247);(1,2,4,3,5,7,6,8,9)(10,12,21)(11,17,39)(13,22,51)(14,24,53)(15,25,52)(16,34,77)(18,40,88)(19,42,89)(20,43,90)(23,59,33)(26,54,107)(27,55,108)(28,56,110)(29,57,111)(30,58,109)(31,61,113)(32,62,112)(35,78,145)(36,79,147)(37,80,148)(38,81,146)(41,95,48)(44,91,158)(45,92,159)(46,93,161)(47,94,160)(49,97,162)(50,98,163)(60,126,76)(63,114,177)(64,115,178)(65,116,180)(66,117,179)(67,118,181)(68,119,182)(69,121,183)(70,122,184)(71,120,186)(72,123,187)(73,124,188)(74,125,185)(75,128,189)(82,149,208)(83,150,210)(84,151,209)(85,152,212)(86,153,213)(87,154,211)(96,170,105)(99,164,217)(100,165,219)(101,166,218)(102,167,220)(103,169,221)(104,168,222)(106,172,223)(127,205,144)(129,190,229)(130,191,228)(131,192,230)(132,194,231)(133,193,233)(134,195,234)(135,196,232)(136,197,235)(137,199,236)(138,201,237)(139,202,238)(140,198,240)(141,200,241)(142,203,242)(143,204,239)(155,214,245)(156,215,247)(157,216,246)(171,227,176)(173,224,248)(174,226,249)(175,225,250)(206,243,251)(207,244,252);(10,13,27)(11,18,45,19,47,103,48,46,104)(12,22,55)(14,29,70)(15,30,69)(16,35,82,36,83,131,38,84,155)(17,40,92,42,94,169,41,93,168)(20,26,64,49,66,132,105,65,133)(21,51,108)(23,56,120)(24,57,122)(25,58,121)(28,71,33)(31,73,139)(32,74,138)(34,78,149,79,150,192,81,151,214)(37,44,99,85,100,136,87,101,173)(39,88,159,89,160,221,95,161,222)(43,54,115,97,117,194,96,116,193)(50,67,68,106,135,137,176,134,140)(52,109,183)(53,111,184)(59,110,186)(60,123,200)(61,124,202)(62,125,201)(63,102,156,129,175,157,130,174,86)(72,141,76)(75,143,207)(77,145,208,147,210,230,146,209,245)(80,91,164,152,165,197,154,166,224)(90,107,178,162,179,231,170,180,233)(98,118,119,172,196,199,171,195,198)(112,185,237)(113,188,238)(114,167,215,190,225,216,191,226,153)(126,187,241)(127,203,243)(128,204,244)(142,206,144)(148,158,217,212,219,235,211,218,248)(163,181,182,223,232,236,227,234,240)(177,220,247,229,250,246,228,249,213)(189,239,252)(205,242,251);(12,23,25)(17,41,42)(21,52,59)(22,56,58)(24,60,62)(34,79,81)(39,89,95)(40,93,94)(43,96,97)(51,109,110)(53,112,126)(54,116,117)(55,120,121)(57,123,125)(61,127,128)(77,146,147)(78,150,151)(80,152,154)(88,160,161)(90,162,170)(91,165,166)(92,168,169)(98,171,172)(107,179,180)(108,183,186)(111,185,187)(113,189,205)(114,190,191)(115,193,194)(118,195,196)(119,198,199)(122,200,201)(124,203,204)(145,209,210)(148,211,212)(149,192,214)(153,215,216)(158,218,219)(159,221,222)(163,223,227)(164,197,224)(167,225,226)(177,228,229)(178,231,233)(181,232,234)(182,236,240)(184,237,241)(188,239,242)(202,243,244)(208,245,230)(213,246,247)(217,248,235)(220,249,250)(238,252,251)Polycyclic presentation:
3^9|a^3=e,b^3=f,c^3=g,h^3=i^2,c^a=ch,d^b=di,e^c=ei,g^a=gi^22691 conjugacy classes. The character table is too large to display.
| maximal subgroup of | H(5,ℤ/9) |
| maximal normal subgroup of | H(5,ℤ/9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 8, F.9^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);