Finite Group DatabasebetaFinite Group Databasebeta| order | 22542 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C13⋊C3)⋊D289 |
| unambiguous semidirect product | C289⋊(D13⋊C3) |
| unambiguous semidirect product | D3757⋊C3 |
| Fitting, derived, Hall | C13×C289 ⋊ C2×C3 |
| Hall | C13⋊C3×C289 ⋊ C2 |
| Hall | D3757 ⋊ C3 |
| Sylow | C289 ⋊ D13⋊C3 |
| Hall | C13⋊C3 ⋊ D289 |
| Sylow | C13 ⋊ C3×D289 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C13×C289 | C2×C3 |
| Frattini | C17 | C17⋊(D13⋊C3) |
| socle | C13×C17 | C3×D17 |
| LCS | G ▷ C13×C289 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 302 |
| transitive degree | μt(G) = 3757 |
A minimal faithful permutation representation — 2 generators on 302 points:
(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,20)(12,23)(14,25)(17,29)(18,28)(19,27)(21,34)(22,37)(24,39)(26,41)(30,46)(31,45)(32,44)(33,43)(35,52)(36,55)(38,57)(40,59)(42,61)(47,67)(48,66)(49,65)(50,64)(51,63)(53,74)(54,77)(56,79)(58,81)(60,83)(62,85)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(75,100)(76,103)(78,105)(80,107)(82,109)(84,111)(86,113)(93,121)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(101,130)(102,133)(104,135)(106,137)(108,139)(110,141)(112,143)(114,145)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(131,132)(134,164)(136,166)(138,168)(140,170)(142,172)(144,174)(146,176)(155,177)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)(163,192)(165,194)(167,196)(169,198)(171,200)(173,202)(175,204)(185,205)(186,211)(187,210)(188,209)(189,208)(190,207)(191,206)(193,218)(195,220)(197,222)(199,224)(201,226)(203,228)(212,229)(213,234)(214,233)(215,232)(216,231)(217,230)(219,240)(221,242)(223,244)(225,246)(227,248)(235,249)(236,253)(237,252)(238,251)(239,250)(241,258)(243,260)(245,262)(247,264)(254,265)(255,268)(256,267)(257,266)(259,272)(261,274)(263,276)(269,277)(270,279)(271,278)(273,282)(275,284)(280,285)(281,286)(283,288)(287,289)(291,295,296,292,293,297)(294,300,301,298,299,302);(1,66,205,110,36,149,268,243,163,218,274,237,98,79,172,156,31,7,91,175,82,21,181,277,221,132,240,284,215,73,107,202,124,18,16,120,144,58,63,209,263,195,101,258,280,189,52,139,228,95,9,29,153,112,38,88,233,245,165,178,272,270,160,77,170,212,70,4,46,177,84,22,117,253,223,134,206,282,256,128,105,200,186,49,13,67,146,60,11,150,265,197,102,230,288,238,99,137,226,157,32,25,92,114,40,43,182,247,167,75,250,287,216,74,168,248,125,19,41,121,86,24,64,210,225,136,147,266,281,190,103,198,235,96,10,61,154,62,12,89,234,199,104,179,278,271,161,135,224,213,71,23,85,122,42,5,118,249,169,76,207,286,257,129,166,246,187,50,39,113,93,26,27,151,227,138,53,231,289,239,100,196,264,158,33,59,145,68,14,44,183,201,106,115,251,283,217,133,222,254,126,20,83,176,47,6,65,211,171,78,148,267,273,191,164,244,236,97,37,111,155,30,2,90,229,140,54,180,279,259,162,194,262,214,72,57,143,123,17,15,119,203,108,35,208,285,241,130,220,276,188,51,81,174,94,8,28,152,173,80,87,232,275,219,131,242,269,159,34,109,204,69,3,45,184,142,56,116,252,261,193,192,260,255,127,55,141,185,48)(290,291,294,296,295,299,301,302,300,293,297,298,292)Polycyclic presentation:
2,3,13,17^2|d^17=e^16,c^a=c^12,d^a=d^16e,e^a=e^16,c^b=c^91016 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22542, 4);