Finite Group DatabasebetaFinite Group Databasebeta| order | 20032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C313 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #14578818 |
| commutator, hypocenter | C313 | C2×C32 |
| Frattini | C4 | C2×#14578818 |
| Fitting | C8×C313 | C8 |
| socle | C2×C313 | C2×C16 |
| prime radical | C2×D313 | C16 |
| cosocle | #19097564 | C22 |
| LCS | G ▷ C313 | |
| UCS | C1 ◁ C8 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 345 |
| transitive degree | μt(G) = 10016 |
A minimal faithful permutation representation — 2 generators on 345 points:
(1,2)(3,17)(4,6)(5,23)(7,9)(8,37)(10,11)(12,62)(13,15)(14,68)(16,18)(19,87)(20,22)(21,92)(24,108)(25,27)(26,53)(28,30)(29,120)(31,32)(33,142)(34,36)(35,147)(38,82)(39,152)(40,42)(41,80)(43,45)(44,79)(46,47)(48,180)(49,51)(50,185)(52,109)(54,56)(55,105)(57,206)(58,60)(59,134)(61,63)(64,217)(65,67)(66,221)(69,226)(70,72)(71,107)(73,75)(74,123)(76,77)(78,81)(83,103)(84,86)(85,214)(88,252)(89,91)(90,174)(93,164)(94,96)(95,155)(97,99)(98,213)(100,101)(102,183)(104,268)(106,227)(110,118)(111,113)(112,251)(114,116)(115,223)(117,172)(119,228)(121,162)(122,232)(124,126)(125,153)(127,128)(129,186)(130,132)(131,163)(133,207)(135,137)(136,196)(138,208)(139,141)(140,225)(143,291)(144,146)(145,294)(148,281)(149,151)(150,159)(154,258)(156,157)(158,273)(160,241)(161,215)(165,256)(166,201)(167,169)(168,195)(170,253)(171,173)(175,242)(176,181)(177,179)(178,209)(182,184)(187,189)(188,234)(190,192)(191,216)(193,194)(197,296)(198,200)(199,246)(202,218)(203,205)(204,255)(210,212)(211,219)(220,257)(222,269)(224,290)(229,250)(230,231)(233,311)(235,270)(236,272)(237,304)(238,240)(239,248)(243,307)(244,245)(247,312)(249,298)(259,299)(260,279)(261,271)(262,278)(263,282)(264,310)(265,267)(266,295)(274,305)(275,277)(276,302)(280,306)(283,297)(284,286)(285,293)(287,289)(288,303)(292,313)(300,301)(308,309);(2,3,5,8,4,7,10,16)(6,12,14,29,13,28,31,61)(9,19,21,44,20,43,46,37)(11,24,26,55,25,54,23,52)(15,33,35,74,34,73,76,63)(17,38,39,18,36,41,78,40)(22,48,50,98,49,97,100,120)(27,57,59,115,58,114,68,133)(30,64,66,125,65,124,127,79)(32,69,71,136,70,135,53,106)(42,83,85,161,84,62,121,122)(45,88,90,168,89,167,92,105)(47,93,95,175,94,108,80,154)(51,102,104,191,103,190,193,123)(56,110,112,156,111,87,166,81)(60,117,119,211,118,210,147,137)(67,129,131,222,130,216,132,213)(72,138,140,228,139,141,134,224)(75,143,145,200,144,230,232,153)(77,148,150,236,149,207,107,197)(82,158,160,109,159,199,152,198)(86,163,165,184,164,142,229,128)(91,170,172,256,171,179,185,223)(96,176,178,259,177,206,214,280)(99,181,183,261,182,205,221,195)(101,186,188,262,187,226,155,243)(113,202,204,278,203,180,260,215)(116,208,209,244,189,217,263,157)(126,218,220,268,219,173,174,196)(146,233,235,281,234,295,194,269)(151,237,239,297,238,290,225,240)(162,247,249,227,248,288,241,287)(169,253,255,299,254,252,251,242)(192,264,266,289,265,308,294,271)(201,274,276,258,275,273,246,301)(212,283,285,300,284,291,303,245)(231,292,277,310,293,306,257,272)(250,267,305,296,304,282,298,270)(279,311,309,307,286,312,302,313)(314,315,316,320,317,321,324,330,318,322,325,331,327,333,336,340,319,323,326,332,328,334,337,341,329,335,338,342,339,343,344,345)Polycyclic presentation:
2^6,313|a^2=c,c^2=d,d^2=e,e^2=f,g^a=g^5,g^b=g^312,g^c=g^25,g^d=g^312376 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20032, 1656);