Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×Q256 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C79 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C79 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C4×C79 ◁ C8×C79 ◁ C16×C79 ◁ C32×C79 ◁ C64×C79 ◁ G | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 335 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 2 generators on 335 points:
(1,2,9,16)(3,17,23,78)(4,24,31,74)(5,32,37,122)(6,38,42,137)(7,43,46,144)(8,47,45,15)(10,49,54,18)(11,55,60,27)(12,61,65,26)(13,66,69,34)(14,70,72,40)(19,82,85,206)(20,86,88,212)(21,89,90,214)(22,91,48,77)(25,62,100,183)(28,108,112,199)(29,113,116,202)(30,117,115,95)(33,56,126,168)(35,129,132,242)(36,133,131,121)(39,75,140,201)(41,141,127,136)(44,76,146,203)(50,153,156,97)(51,157,159,96)(52,160,161,145)(53,81,73,149)(57,169,172,223)(58,173,175,222)(59,107,147,165)(63,184,186,225)(64,104,148,179)(67,151,191,79)(68,128,135,188)(71,152,194,80)(83,207,208,252)(84,209,150,205)(87,213,138,211)(92,163,196,101)(93,162,197,105)(94,193,198,110)(98,185,221,256)(99,187,118,182)(102,120,177,239)(103,119,178,238)(106,143,164,244)(109,180,232,254)(111,233,226,229)(114,181,235,255)(123,170,240,250)(124,174,241,253)(125,176,134,167)(130,166,243,248)(139,210,142,200)(154,234,245,231)(155,220,195,230)(158,219,189,237)(171,251,215,249)(190,204,192,218)(216,224,236,247)(217,227,228,246)(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);(1,3,9,23)(2,10,16,54)(4,25,31,100)(5,33,37,126)(6,39,42,140)(7,44,46,146)(8,48,45,22)(11,56,60,168)(12,62,65,183)(13,67,69,191)(14,71,72,194)(15,73,47,53)(17,74,78,24)(18,55,49,27)(19,61,85,26)(20,66,88,34)(21,70,90,40)(28,109,112,232)(29,114,116,235)(30,118,115,99)(32,50,122,156)(35,130,132,243)(36,134,131,125)(38,51,137,159)(41,142,127,139)(43,52,144,161)(57,170,172,250)(58,174,175,253)(59,176,147,167)(63,185,186,256)(64,187,148,182)(68,192,135,190)(75,199,201,108)(76,202,203,113)(77,95,91,117)(79,169,151,223)(80,173,152,222)(81,107,149,165)(82,196,206,92)(83,184,208,225)(84,104,150,179)(86,197,212,93)(87,128,138,188)(89,198,214,94)(96,163,157,101)(97,162,153,105)(98,193,221,110)(102,124,177,241)(103,123,178,240)(106,145,164,160)(111,234,226,231)(119,181,238,255)(120,180,239,254)(121,195,133,155)(129,154,242,245)(136,189,141,158)(143,166,244,248)(171,252,215,207)(200,229,210,233)(204,251,218,249)(205,216,209,236)(211,217,213,228)(219,224,237,247)(220,227,230,246)5293 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 541);