Finite Group DatabasebetaFinite Group Databasebeta| order | 13851 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C19×C27 · Schur multiplier (H2) = C32 |
| direct product | C19×#10624833 |
| subgroup | G/H | |
|---|---|---|
| center | C19×C81 | C32 |
| commutator | C3 | C32×C19×C27 |
| Frattini, cosocle | C27 | C33×C19 |
| socle | C3×C19 | C32×C27 |
| prime radical | C19×H(3,𝔽3) | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C19×C81 ◁ G | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 262 |
| transitive degree | μt(G) = 4617 |
A minimal faithful permutation representation — 3 generators on 262 points:
(1,2,7,4,9,24,17,32,61,5,10,25,18,33,62,49,78,117,20,35,64,51,80,119,100,139,176,6,11,26,19,34,63,50,79,118,21,36,65,52,81,120,101,140,177,54,83,122,103,142,179,158,195,219,22,37,66,53,82,121,102,141,178,55,84,123,104,143,180,159,196,220,105,144,181,160,197,221,205,229,239)(3,8,23,14,29,58,44,73,112,15,30,59,45,74,113,95,134,171,47,76,115,97,136,173,155,192,216,16,31,60,46,75,114,96,135,172,48,77,116,98,137,174,156,193,217,99,138,175,157,194,218,204,228,238,13,28,57,42,71,110,92,131,168,43,72,111,93,132,169,153,190,214,94,133,170,154,191,215,203,227,237)(12,27,56,39,68,107,87,126,163,40,69,108,88,127,164,148,185,209,89,128,165,149,186,210,200,224,234,38,67,106,85,124,161,145,182,206,86,125,162,146,183,207,198,222,232,147,184,208,199,223,233,230,240,242,41,70,109,90,129,166,150,187,211,91,130,167,151,188,212,201,225,235,152,189,213,202,226,236,231,241,243)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262);(1,3,12)(2,8,27)(4,14,39)(5,15,40)(6,16,38)(7,23,56)(9,29,68)(10,30,69)(11,31,67)(13,41,22)(17,44,87)(18,45,88)(19,46,85)(20,47,89)(21,48,86)(24,58,107)(25,59,108)(26,60,106)(28,70,37)(32,73,126)(33,74,127)(34,75,124)(35,76,128)(36,77,125)(42,90,53)(43,91,55)(49,95,148)(50,96,145)(51,97,149)(52,98,146)(54,99,147)(57,109,66)(61,112,163)(62,113,164)(63,114,161)(64,115,165)(65,116,162)(71,129,82)(72,130,84)(78,134,185)(79,135,182)(80,136,186)(81,137,183)(83,138,184)(92,150,102)(93,151,104)(94,152,105)(100,155,200)(101,156,198)(103,157,199)(110,166,121)(111,167,123)(117,171,209)(118,172,206)(119,173,210)(120,174,207)(122,175,208)(131,187,141)(132,188,143)(133,189,144)(139,192,224)(140,193,222)(142,194,223)(153,201,159)(154,202,160)(158,204,230)(168,211,178)(169,212,180)(170,213,181)(176,216,234)(177,217,232)(179,218,233)(190,225,196)(191,226,197)(195,228,240)(203,231,205)(214,235,220)(215,236,221)(219,238,242)(227,241,229)(237,243,239);(3,13,16)(8,28,31)(12,38,41)(14,42,46)(15,43,48)(23,57,60)(27,67,70)(29,71,75)(30,72,77)(39,85,90)(40,86,91)(44,92,96)(45,93,98)(47,94,99)(56,106,109)(58,110,114)(59,111,116)(68,124,129)(69,125,130)(73,131,135)(74,132,137)(76,133,138)(87,145,150)(88,146,151)(89,147,152)(95,153,156)(97,154,157)(107,161,166)(108,162,167)(112,168,172)(113,169,174)(115,170,175)(126,182,187)(127,183,188)(128,184,189)(134,190,193)(136,191,194)(148,198,201)(149,199,202)(155,203,204)(163,206,211)(164,207,212)(165,208,213)(171,214,217)(173,215,218)(185,222,225)(186,223,226)(192,227,228)(200,230,231)(209,232,235)(210,233,236)(216,237,238)(224,240,241)(234,242,243)5643 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13851, 393);