Finite Group DatabasebetaFinite Group Databasebeta| order | 20050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C25 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #23756633⋊C2 |
| derived, Sylow | C401 ⋊ C2×C25 |
| Hall | #23756633 ⋊ C2 |
| Hall | D401 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C5 | D401⋊C5 |
| commutator, hypocenter | C401 | C2×C25 |
| Fitting, socle, cosocle | C5×C401 | C2×C5 |
| prime radical | C5×D401 | C5 |
| LCS | G ▷ C401 | |
| UCS | C1 ◁ C5 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 426 |
| transitive degree | μt(G) = 10025 |
A minimal faithful permutation representation — 2 generators on 426 points:
(2,6,7,25,30,3,4,12,14,54)(5,19,20,87,92,9,10,41,43,163)(8,35,36,37,65,16,17,75,77,79)(11,48,49,192,157,22,23,103,105,257)(13,59,60,215,204,27,28,125,127,115)(15,69,70,66,67,32,33,71,31,78)(18,81,82,224,173,38,39,153,155,233)(21,97,98,53,137,45,46,183,112,113)(24,110,111,64,121,51,52,83,135,136)(26,119,120,55,206,56,57,209,116,117)(29,132,133,248,162,62,63,148,229,264)(34,143,144,174,175,72,73,239,93,281)(40,158,159,359,258,84,85,259,260,357)(42,168,169,372,320,89,90,271,273,205)(44,177,178,138,139,94,95,285,68,109)(47,106,188,193,232,100,101,197,302,140)(50,99,199,298,149,107,108,186,207,76)(58,134,213,164,114,122,123,230,265,202)(61,222,223,80,214,129,130,282,151,124)(74,245,246,312,263,146,147,346,196,364)(86,171,172,354,212,161,91,278,279,325)(88,267,268,118,314,165,166,367,198,208)(96,291,292,283,284,180,181,390,176,382)(102,304,191,360,358,189,190,305,306,399)(104,309,310,378,395,194,195,392,393,321)(126,241,288,289,250,217,218,179,220,150)(128,287,236,237,255,182,221,142,210,156)(131,280,333,332,203,225,226,379,374,318)(141,242,243,295,296,234,145,216,185,211)(152,334,275,276,365,252,227,170,361,377)(154,160,337,363,369,201,254,262,269,322)(167,341,370,266,324,240,270,400,362,366)(184,247,249,315,368,235,294,349,351,323)(187,387,343,344,383,299,300,244,331,303)(200,272,394,277,398,316,317,373,347,330)(219,371,345,380,307,328,329,401,336,381)(228,348,340,311,353,335,253,313,375,251)(231,261,274,308,319,338,339,356,376,238)(256,342,391,388,389,355,301,293,290,297)(286,352,396,397,386,384,385,327,350,326)(402,406,414,422,426,404,410,419,425,412,407,415,423,408,417,411,420,405,413,421,416,403,409,418,424);(1,2,5,11,24,53,114,203,319,236,70,69,141,235,46,99,187,301,218,239,292,391,291,385,349,133,146,72,32,12,26,58,124,117,207,322,395,320,204,30,65,137,149,17,37,80,152,253,346,178,287,387,396,243,60,128,180,94,41,88,167,73,145,244,345,394,366,296,209,135,62,27,4,8,18,40,86,162,92,173,232,138,66,25,55,118,101,103,193,308,329,341,388,288,177,286,386,230,337,328,217,28,61,131,227,208,323,367,279,353,250,67,75,110,201,39,83,157,258,358,398,289,215,199,315,344,130,224,332,317,90,170,277,147,247,350,400,169,274,355,255,153,171,89,10,21,47,102,191,307,175,282,105,196,237,57,121,212,190,195,311,372,363,267,283,174,87,164,266,339,259,360,335,63,134,231,340,188,303,382,379,392,316,139,107,48,104,85,160,263,365,260,361,389,295,183,229,336,226,278,378,393,172,280,380,248,98,185,297,276,359,377,364,254,158,194,23,50,109,200,310,333,284,383,197,313,338,123,132,228,306,159,261,362,265,43,93,176,166,269,273,375,309,304,325,136,119,210,312,192,223,281,381,305,189,100,45,19,42,91,82,156,256,356,271,370,397,294,245,347,275,168,272,374,155,222,331,351,186,127,220,330,399,357,257,111,81,154,52,36,78,150,251,354,268,368,314,334,225,129,59,126,219,262,213,326,384,95,179,290,270,371,376,302,49,106,198,116,14,31,68,140,233,163,264,161,84,38,16,6,13,29,64,120,211,324,373,401,343,242,143,240,165,20,44,96,182,125,216,327,300,221,285,246,348,252,151,77,35,76,113,79,54,115,205,321,369,298,206,214,122,56,7,15,34,74,148,249,352,181,293,390,144,241,342,299,108,97,184,234,33,71,142,238,318,202,112,51,22,9,3)Polycyclic presentation:
2,5,401,5|b^5=d,c^a=c^400,c^b=c^72250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20050, 2);