Finite Group DatabasebetaFinite Group Databasebeta| order | 20032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊C4×C313 ⋊ C4 |
| Sylow | C313 ⋊ #385 |
234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 321 |
| transitive degree | μt(G) = 2504 |
A minimal faithful permutation representation — 3 generators on 321 points:
(1,2,6,12,22,36,54,76,102,132,166,204,246,289,293,248,206,168,134,104,78,56,38,24,14,4,8,16,28,42,60,82,108,138,172,210,252,292,243,287,297,254,212,174,140,110,84,62,44,26,10,18,30,46,66,88,114,144,178,216,258,296,247,201,241,285,301,260,218,180,146,116,90,64,40,20,32,48,68,92,120,150,184,222,264,300,253,205,163,199,239,283,305,266,224,186,152,118,86,58,34,50,70,94,122,154,190,228,270,304,259,211,167,129,161,197,237,281,309,272,230,188,148,112,80,52,72,96,124,156,192,232,276,308,265,217,173,133,99,127,159,195,235,279,313,274,226,182,142,106,74,98,126,158,194,234,278,312,271,223,179,139,103,73,97,125,157,193,233,277,310,268,220,176,136,100,128,160,196,236,280,311,275,229,185,145,109,77,51,71,95,123,155,191,231,273,306,262,214,170,130,162,198,238,282,307,269,227,189,151,115,83,55,33,49,69,93,121,153,187,225,267,302,256,208,164,200,240,284,303,263,221,183,149,119,89,61,37,19,31,47,67,91,117,147,181,219,261,298,250,202,242,286,299,257,215,177,143,113,87,65,43,23,9,17,29,45,63,85,111,141,175,213,255,294,244,288,295,251,209,171,137,107,81,59,41,27,13,3,7,15,25,39,57,79,105,135,169,207,249,290,291,245,203,165,131,101,75,53,35,21,11,5)(319,321);(2,4,5,3)(6,10,11,9)(7,8,14,13)(12,20,21,19)(15,18,24,23)(16,26,27,17)(22,34,35,33)(25,32,38,37)(28,40,41,31)(29,30,44,43)(36,52,53,51)(39,50,56,55)(42,58,59,49)(45,48,62,61)(46,64,65,47)(54,74,75,73)(57,72,78,77)(60,80,81,71)(63,70,84,83)(66,86,87,69)(67,68,90,89)(76,100,101,99)(79,98,104,103)(82,106,107,97)(85,96,110,109)(88,112,113,95)(91,94,116,115)(92,118,119,93)(102,130,131,129)(105,128,134,133)(108,136,137,127)(111,126,140,139)(114,142,143,125)(117,124,146,145)(120,148,149,123)(121,122,152,151)(132,164,165,163)(135,162,168,167)(138,170,171,161)(141,160,174,173)(144,176,177,159)(147,158,180,179)(150,182,183,157)(153,156,186,185)(154,188,189,155)(166,202,203,201)(169,200,206,205)(172,208,209,199)(175,198,212,211)(178,214,215,197)(181,196,218,217)(184,220,221,195)(187,194,224,223)(190,226,227,193)(191,192,230,229)(204,244,245,243)(207,242,248,247)(210,250,251,241)(213,240,254,253)(216,256,257,239)(219,238,260,259)(222,262,263,237)(225,236,266,265)(228,268,269,235)(231,234,272,271)(232,274,275,233)(246,290,291,289)(249,288,293,292)(252,294,295,287)(255,286,297,296)(258,298,299,285)(261,284,301,300)(264,302,303,283)(267,282,305,304)(270,306,307,281)(273,280,309,308)(276,310,311,279)(277,278,313,312)(315,317);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,83,84,70)(66,69,87,86)(67,89,90,68)(76,99,101,100)(79,103,104,98)(82,97,107,106)(85,109,110,96)(88,95,113,112)(91,115,116,94)(92,93,119,118)(102,129,131,130)(105,133,134,128)(108,127,137,136)(111,139,140,126)(114,125,143,142)(117,145,146,124)(120,123,149,148)(121,151,152,122)(132,163,165,164)(135,167,168,162)(138,161,171,170)(141,173,174,160)(144,159,177,176)(147,179,180,158)(150,157,183,182)(153,185,186,156)(154,155,189,188)(166,201,203,202)(169,205,206,200)(172,199,209,208)(175,211,212,198)(178,197,215,214)(181,217,218,196)(184,195,221,220)(187,223,224,194)(190,193,227,226)(191,229,230,192)(204,243,245,244)(207,247,248,242)(210,241,251,250)(213,253,254,240)(216,239,257,256)(219,259,260,238)(222,237,263,262)(225,265,266,236)(228,235,269,268)(231,271,272,234)(232,233,275,274)(246,289,291,290)(249,292,293,288)(252,287,295,294)(255,296,297,286)(258,285,299,298)(261,300,301,284)(264,283,303,302)(267,304,305,282)(270,281,307,306)(273,308,309,280)(276,279,311,310)(277,312,313,278)(314,315)(316,317)(318,319)(320,321)Polycyclic presentation:
2^6,313|a^2=f,b^2=f,b^a=bd,c^a=ce,g^a=g^288,g^b=g^25,g^f=g^312808 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20032, 1496);