Finite Group DatabasebetaFinite Group Databasebeta| order | 36224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D18112 |
| Fitting | C64×C283 ⋊ C2 |
| Sylow | C283 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D9056 |
| commutator, cosocle | C32×C283 | C22 |
| Frattini | C32 | C2×D283 |
| Fitting | C64×C283 | C2 |
| socle | C2×C283 | D32 |
| hypercenter | C64 | D283 |
| hypocenter | C283 | D64 |
| LCS | G ▷ C32×C283 ▷ C16×C283 ▷ C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 347 |
| transitive degree | μt(G) = 18112 |
A minimal faithful permutation representation — 2 generators on 347 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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,347)(67,346)(68,345)(69,344)(70,343)(71,342)(72,341)(73,340)(74,339)(75,338)(76,337)(77,336)(78,335)(79,334)(80,333)(81,332)(82,331)(83,330)(84,329)(85,328)(86,327)(87,326)(88,325)(89,324)(90,323)(91,322)(92,321)(93,320)(94,319)(95,318)(96,317)(97,316)(98,315)(99,314)(100,313)(101,312)(102,311)(103,310)(104,309)(105,308)(106,307)(107,306)(108,305)(109,304)(110,303)(111,302)(112,301)(113,300)(114,299)(115,298)(116,297)(117,296)(118,295)(119,294)(120,293)(121,292)(122,291)(123,290)(124,289)(125,288)(126,287)(127,286)(128,285)(129,284)(130,283)(131,282)(132,281)(133,280)(134,279)(135,278)(136,277)(137,276)(138,275)(139,274)(140,273)(141,272)(142,271)(143,270)(144,269)(145,268)(146,267)(147,266)(148,265)(149,264)(150,263)(151,262)(152,261)(153,260)(154,259)(155,258)(156,257)(157,256)(158,255)(159,254)(160,253)(161,252)(162,251)(163,250)(164,249)(165,248)(166,247)(167,246)(168,245)(169,244)(170,243)(171,242)(172,241)(173,240)(174,239)(175,238)(176,237)(177,236)(178,235)(179,234)(180,233)(181,232)(182,231)(183,230)(184,229)(185,228)(186,227)(187,226)(188,225)(189,224)(190,223)(191,222)(192,221)(193,220)(194,219)(195,218)(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)Polycyclic presentation:
2^7,283|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^282,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2829059 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36224, 2723);