Finite Group DatabasebetaFinite Group Databasebeta| order | 51302 |
| automorphism | Aut(G) order 649278112 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D25651 |
| Fitting, socle, derived, Hall | C113×C227 ⋊ C2 |
| Sylow | C227 ⋊ D113 |
| Sylow | C113 ⋊ D227 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C113×C227 | C2 |
| LCS | G ▷ C113×C227 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 340 |
| transitive degree | μt(G) = 25651 |
A minimal faithful permutation representation — 2 generators on 340 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)(114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340);(2,113)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(115,340)(116,339)(117,338)(118,337)(119,336)(120,335)(121,334)(122,333)(123,332)(124,331)(125,330)(126,329)(127,328)(128,327)(129,326)(130,325)(131,324)(132,323)(133,322)(134,321)(135,320)(136,319)(137,318)(138,317)(139,316)(140,315)(141,314)(142,313)(143,312)(144,311)(145,310)(146,309)(147,308)(148,307)(149,306)(150,305)(151,304)(152,303)(153,302)(154,301)(155,300)(156,299)(157,298)(158,297)(159,296)(160,295)(161,294)(162,293)(163,292)(164,291)(165,290)(166,289)(167,288)(168,287)(169,286)(170,285)(171,284)(172,283)(173,282)(174,281)(175,280)(176,279)(177,278)(178,277)(179,276)(180,275)(181,274)(182,273)(183,272)(184,271)(185,270)(186,269)(187,268)(188,267)(189,266)(190,265)(191,264)(192,263)(193,262)(194,261)(195,260)(196,259)(197,258)(198,257)(199,256)(200,255)(201,254)(202,253)(203,252)(204,251)(205,250)(206,249)(207,248)(208,247)(209,246)(210,245)(211,244)(212,243)(213,242)(214,241)(215,240)(216,239)(217,238)(218,237)(219,236)(220,235)(221,234)(222,233)(223,232)(224,231)(225,230)(226,229)(227,228)Polycyclic presentation:
2,113,227|b^a=b^112,c^a=c^22612827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(51302, 5);