Finite Group DatabasebetaFinite Group Databasebeta| order | 26286 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C337⋊C3)⋊D13 |
| Fitting, socle, derived, Hall | C13×C337 ⋊ C2×C3 |
| Hall | C13×C337⋊C3 ⋊ C2 |
| Hall | D4381 ⋊ C3 |
| Hall | C337⋊C3 ⋊ D13 |
| Sylow | C337 ⋊ C3×D13 |
| Sylow | C13 ⋊ D337⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C13×C337 | C2×C3 |
| LCS | G ▷ C13×C337 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 350 |
| transitive degree | μt(G) = 4381 |
A minimal faithful permutation representation — 2 generators on 350 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,317)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,285,286,236,237,262)(198,292,293,243,244,257)(203,299,300,249,250,252)(208,304,297,254,255,247)(212,307,290,259,260,240)(215,310,283,263,264,233)(226,313,284,268,277,324)(228,320,321,278,276,314)(231,270,319,280,281,315)(238,326,327,287,288,312)(245,330,331,294,295,309)(251,334,335,301,302,306)(256,332,336,296,305,303)(261,328,333,289,308,298)(265,322,329,279,311,291)(282,318,325,316,323,337)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350);(1,2,5,11,19,32,51,74,103,138,179,226,230,137,176,190,197,202,207,211,214,216,266,312,336,300,242,188,132,70,95,117,111,96,69,94,127,165,208,256,294,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,291,275,319,277,322,328,332,334,330,326,320,274,318,337,324,283,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,289,259,210,142,73,100,114,119,123,126,128,166,209,257,306,331,286,220,134,171,205,200,194,187,172,133,170,215,265,278,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,282,281,178,223,237,244,250,255,260,264,267,269,314,333,297,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,301,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,298,321,222,271,310,307,304,299,292,285,272,221,270,316,280,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,279,263,183,102,135,149,155,160,164,167,169,213,262,309,335,293,235,173,99,130,158,153,146,131,98,129,168,212,261,287,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,284,325,323,225,276,288,295,302,305,308,311,313,315,317,329,290,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,296,254,206,163,107,50,71,84,88,91,93,125,162,204,252,303,327,273,175,218,258,253,248,241,234,219,174,217,268,224,177,136,101,72,49,30,17,9,3)(338,339,341,343,345,347,349,350,348,346,344,342,340)Polycyclic presentation:
2,3,13,337|c^a=c^12,d^a=d^336,d^b=d^128752 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26286, 21);