Finite Group DatabasebetaFinite Group Databasebeta| order | 34432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D17216 |
| Fitting | C64×C269 ⋊ C2 |
| Sylow | C269 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8608 |
| commutator, cosocle | C32×C269 | C22 |
| Frattini | C32 | C2×D269 |
| Fitting | C64×C269 | C2 |
| socle | C2×C269 | D32 |
| hypercenter | C64 | D269 |
| hypocenter | C269 | D64 |
| LCS | G ▷ C32×C269 ▷ C16×C269 ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 333 |
| transitive degree | μt(G) = 17216 |
A minimal faithful permutation representation — 2 generators on 333 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);(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)(66,333)(67,332)(68,331)(69,330)(70,329)(71,328)(72,327)(73,326)(74,325)(75,324)(76,323)(77,322)(78,321)(79,320)(80,319)(81,318)(82,317)(83,316)(84,315)(85,314)(86,313)(87,312)(88,311)(89,310)(90,309)(91,308)(92,307)(93,306)(94,305)(95,304)(96,303)(97,302)(98,301)(99,300)(100,299)(101,298)(102,297)(103,296)(104,295)(105,294)(106,293)(107,292)(108,291)(109,290)(110,289)(111,288)(112,287)(113,286)(114,285)(115,284)(116,283)(117,282)(118,281)(119,280)(120,279)(121,278)(122,277)(123,276)(124,275)(125,274)(126,273)(127,272)(128,271)(129,270)(130,269)(131,268)(132,267)(133,266)(134,265)(135,264)(136,263)(137,262)(138,261)(139,260)(140,259)(141,258)(142,257)(143,256)(144,255)(145,254)(146,253)(147,252)(148,251)(149,250)(150,249)(151,248)(152,247)(153,246)(154,245)(155,244)(156,243)(157,242)(158,241)(159,240)(160,239)(161,238)(162,237)(163,236)(164,235)(165,234)(166,233)(167,232)(168,231)(169,230)(170,229)(171,228)(172,227)(173,226)(174,225)(175,224)(176,223)(177,222)(178,221)(179,220)(180,219)(181,218)(182,217)(183,216)(184,215)(185,214)(186,213)(187,212)(188,211)(189,210)(190,209)(191,208)(192,207)(193,206)(194,205)(195,204)(196,203)(197,202)(198,201)(199,200)Polycyclic presentation:
2^7,269|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^268,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2688611 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34432, 2723);