Finite Group DatabasebetaFinite Group Databasebeta| order | 27008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic6752 |
| Sylow | C211 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6752 |
| commutator, cosocle | C32×C211 | C22 |
| Frattini | C32 | C2×D211 |
| Fitting | C64×C211 | C2 |
| socle, prime radical | C2×C211 | D32 |
| hypercenter | C64 | D211 |
| hypocenter | C211 | Q128 |
| LCS | G ▷ C32×C211 ▷ C16×C211 ▷ C8×C211 ▷ C4×C211 ▷ C2×C211 ▷ C211 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 339 |
| transitive degree | μt(G) = 27008 |
A minimal faithful permutation representation — 2 generators on 339 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(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);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,339)(131,338)(132,337)(133,336)(134,335)(135,334)(136,333)(137,332)(138,331)(139,330)(140,329)(141,328)(142,327)(143,326)(144,325)(145,324)(146,323)(147,322)(148,321)(149,320)(150,319)(151,318)(152,317)(153,316)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)(160,309)(161,308)(162,307)(163,306)(164,305)(165,304)(166,303)(167,302)(168,301)(169,300)(170,299)(171,298)(172,297)(173,296)(174,295)(175,294)(176,293)(177,292)(178,291)(179,290)(180,289)(181,288)(182,287)(183,286)(184,285)(185,284)(186,283)(187,282)(188,281)(189,280)(190,279)(191,278)(192,277)(193,276)(194,275)(195,274)(196,273)(197,272)(198,271)(199,270)(200,269)(201,268)(202,267)(203,266)(204,265)(205,264)(206,263)(207,262)(208,261)(209,260)(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)(216,253)(217,252)(218,251)(219,250)(220,249)(221,248)(222,247)(223,246)(224,245)(225,244)(226,243)(227,242)(228,241)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235)Polycyclic presentation:
2^7,211|a^2=g,b^2=g,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^210,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2106755 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27008, 2728);