Finite Group DatabasebetaFinite Group Databasebeta| order | 20050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C25 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #23756635⋊C2 |
| Fitting, socle, derived, Sylow | C401 ⋊ C2×C25 |
| Hall | #23756635 ⋊ C2 |
| Hall | D401 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C401 | C2×C25 |
| prime radical | D401⋊C5 | C5 |
| cosocle | C401⋊C5 | C2×C5 |
| LCS | G ▷ C401 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 401 |
| transitive degree | μt(G) = 401 |
A minimal faithful permutation representation — 2 generators on 401 points:
(2,19,67,116,108,8,26,266,173,126,12,132,157,49,159,58,63,18,65,115,87,7,24,44,171,3,10,128,70,238,5,52,165,133,64,21,71,302,23,263,32,156,9,127,69,147,4,50,25,248)(6,101,56,80,155,39,107,166,89,336,43,361,201,204,140,229,187,100,55,215,175,38,20,213,88,14,41,139,33,300,16,224,86,190,233,103,236,314,105,279,113,337,40,138,281,325,15,54,141,189)(11,182,326,59,239,77,188,96,218,374,81,163,98,293,291,66,282,181,148,334,203,76,186,319,217,28,79,303,97,245,30,332,37,149,335,183,310,216,185,184,162,295,78,170,393,150,29,331,292,137)(13,205,350,371,197,91,207,237,84,397,94,214,31,347,191,95,301,196,328,193,202,90,199,271,395,34,92,313,355,346,36,342,312,195,349,192,164,83,198,82,212,309,85,252,341,161,35,348,354,343)(17,220,364,265,381,118,222,352,256,382,121,200,392,258,73,179,290,250,277,75,363,117,221,369,284,45,99,232,206,257,47,178,289,122,378,74,146,344,249,27,255,353,119,231,177,251,46,104,391,143)(22,144,253,384,324,152,120,267,386,333,109,129,270,280,112,111,131,296,287,273,372,151,145,370,375,60,57,241,269,340,62,110,130,399,264,42,51,385,297,243,240,268,153,318,244,242,61,286,330,388)(48,235,366,102,176,260,383,285,278,154,262,305,283,365,194,158,380,234,311,320,294,106,228,172,307,123,261,247,226,72,125,362,225,227,298,219,376,142,339,299,304,373,259,246,345,306,124,114,208,160)(53,93,368,308,316,275,356,400,168,379,276,359,68,254,389,321,223,315,367,338,323,274,230,396,398,134,209,358,394,329,136,211,360,180,288,390,322,167,317,387,174,272,169,357,351,327,135,210,401,377);(1,2,6,17,48,23,66,164,312,224,105,190,340,321,116,249,248,377,362,188,339,329,227,217,134,10,27,29,80,21,59,150,189,292,369,395,347,250,372,87,202,127,265,36,95,215,297,56,143,285,41,108,239,243,109,38,102,228,103,230,296,182,333,64,160,308,270,388,185,304,156,60,5,13,37,99,223,71,175,326,302,283,138,58,148,167,50,74,165,313,320,145,288,305,366,393,291,30,82,194,261,62,155,301,287,315,390,400,356,40,106,141,192,260,197,98,219,363,46,120,254,380,111,231,275,159,233,331,342,396,232,370,168,177,327,335,253,352,199,351,337,181,274,221,16,44,18,51,61,154,295,279,255,381,236,149,294,322,131,271,314,392,341,358,246,310,229,81,130,85,200,144,54,4,11,31,86,128,69,172,205,355,367,234,256,118,252,222,349,378,386,290,269,387,244,173,26,72,35,93,84,195,209,90,176,52,133,273,389,384,353,399,382,397,178,328,47,122,259,357,371,92,208,115,67,166,83,28,7,20,57,146,196,267,183,113,163,311,368,193,344,201,354,268,359,306,218,361,257,179,140,282,298,151,129,9,25,39,104,135,78,187,338,348,289,241,374,323,75,180,330,364,401,207,186,336,263,136,277,240,373,317,110,117,251,262,216,346,125,94,213,124,100,211,360,276,169,318,309,300,152,235,299,191,77,184,334,247,376,379,286,345,350,266,121,24,68,170,32,55,142,157,303,325,132,272,220,96,34,8,22,63,158,293,375,385,394,307,126,264,79,153,210,43,114,226,15,42,112,245,238,101,225,284,139,280,281,212,91,206,65,161,147,242,119,198,343,391,319,88,203,97,12,33,76,73,19,53,137,278,316,365,332,383,398,171,258,70,174,324,89,204,107,237,214,162,49,123,45,14,3)Polycyclic presentation:
2,5^2,401|b^5=c,d^a=d^400,d^b=d^173,d^c=d^7258 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20050, 8);