Finite Group DatabasebetaFinite Group Databasebeta| order | 37504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D18752 |
| Fitting | C64×C293 ⋊ C2 |
| Sylow | C293 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D9376 |
| commutator, cosocle | C32×C293 | C22 |
| Frattini | C32 | C2×D293 |
| Fitting | C64×C293 | C2 |
| socle | C2×C293 | D32 |
| hypercenter | C64 | D293 |
| hypocenter | C293 | D64 |
| LCS | G ▷ C32×C293 ▷ C16×C293 ▷ C8×C293 ▷ C4×C293 ▷ C2×C293 ▷ C293 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 357 |
| transitive degree | μt(G) = 18752 |
A minimal faithful permutation representation — 2 generators on 357 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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357);(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)(66,357)(67,356)(68,355)(69,354)(70,353)(71,352)(72,351)(73,350)(74,349)(75,348)(76,347)(77,346)(78,345)(79,344)(80,343)(81,342)(82,341)(83,340)(84,339)(85,338)(86,337)(87,336)(88,335)(89,334)(90,333)(91,332)(92,331)(93,330)(94,329)(95,328)(96,327)(97,326)(98,325)(99,324)(100,323)(101,322)(102,321)(103,320)(104,319)(105,318)(106,317)(107,316)(108,315)(109,314)(110,313)(111,312)(112,311)(113,310)(114,309)(115,308)(116,307)(117,306)(118,305)(119,304)(120,303)(121,302)(122,301)(123,300)(124,299)(125,298)(126,297)(127,296)(128,295)(129,294)(130,293)(131,292)(132,291)(133,290)(134,289)(135,288)(136,287)(137,286)(138,285)(139,284)(140,283)(141,282)(142,281)(143,280)(144,279)(145,278)(146,277)(147,276)(148,275)(149,274)(150,273)(151,272)(152,271)(153,270)(154,269)(155,268)(156,267)(157,266)(158,265)(159,264)(160,263)(161,262)(162,261)(163,260)(164,259)(165,258)(166,257)(167,256)(168,255)(169,254)(170,253)(171,252)(172,251)(173,250)(174,249)(175,248)(176,247)(177,246)(178,245)(179,244)(180,243)(181,242)(182,241)(183,240)(184,239)(185,238)(186,237)(187,236)(188,235)(189,234)(190,233)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213)(211,212)Polycyclic presentation:
2^7,293|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^292,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2929379 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37504, 2723);