Finite Group DatabasebetaFinite Group Databasebeta| order | 33664 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D16832 |
| Fitting | C64×C263 ⋊ C2 |
| Sylow | C263 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8416 |
| commutator, cosocle | C32×C263 | C22 |
| Frattini | C32 | C2×D263 |
| Fitting | C64×C263 | C2 |
| socle | C2×C263 | D32 |
| hypercenter | C64 | D263 |
| hypocenter | C263 | D64 |
| LCS | G ▷ C32×C263 ▷ C16×C263 ▷ C8×C263 ▷ C4×C263 ▷ C2×C263 ▷ C263 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 327 |
| transitive degree | μt(G) = 16832 |
A minimal faithful permutation representation — 2 generators on 327 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);(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)(66,327)(67,326)(68,325)(69,324)(70,323)(71,322)(72,321)(73,320)(74,319)(75,318)(76,317)(77,316)(78,315)(79,314)(80,313)(81,312)(82,311)(83,310)(84,309)(85,308)(86,307)(87,306)(88,305)(89,304)(90,303)(91,302)(92,301)(93,300)(94,299)(95,298)(96,297)(97,296)(98,295)(99,294)(100,293)(101,292)(102,291)(103,290)(104,289)(105,288)(106,287)(107,286)(108,285)(109,284)(110,283)(111,282)(112,281)(113,280)(114,279)(115,278)(116,277)(117,276)(118,275)(119,274)(120,273)(121,272)(122,271)(123,270)(124,269)(125,268)(126,267)(127,266)(128,265)(129,264)(130,263)(131,262)(132,261)(133,260)(134,259)(135,258)(136,257)(137,256)(138,255)(139,254)(140,253)(141,252)(142,251)(143,250)(144,249)(145,248)(146,247)(147,246)(148,245)(149,244)(150,243)(151,242)(152,241)(153,240)(154,239)(155,238)(156,237)(157,236)(158,235)(159,234)(160,233)(161,232)(162,231)(163,230)(164,229)(165,228)(166,227)(167,226)(168,225)(169,224)(170,223)(171,222)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,210)(184,209)(185,208)(186,207)(187,206)(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197)Polycyclic presentation:
2^7,263|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^262,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2628419 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33664, 2723);