Finite Group DatabasebetaFinite Group Databasebeta| order | 20822 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D10411 |
| unambiguous semidirect product | C29⋊D359 |
| unambiguous semidirect product | C359⋊D29 |
| Fitting, socle, derived, Hall | C29×C359 ⋊ C2 |
| Sylow | C359 ⋊ D29 |
| Sylow | C29 ⋊ D359 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C29×C359 | C2 |
| LCS | G ▷ C29×C359 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 388 |
| transitive degree | μt(G) = 10411 |
A minimal faithful permutation representation — 2 generators on 388 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388);(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,388)(32,387)(33,386)(34,385)(35,384)(36,383)(37,382)(38,381)(39,380)(40,379)(41,378)(42,377)(43,376)(44,375)(45,374)(46,373)(47,372)(48,371)(49,370)(50,369)(51,368)(52,367)(53,366)(54,365)(55,364)(56,363)(57,362)(58,361)(59,360)(60,359)(61,358)(62,357)(63,356)(64,355)(65,354)(66,353)(67,352)(68,351)(69,350)(70,349)(71,348)(72,347)(73,346)(74,345)(75,344)(76,343)(77,342)(78,341)(79,340)(80,339)(81,338)(82,337)(83,336)(84,335)(85,334)(86,333)(87,332)(88,331)(89,330)(90,329)(91,328)(92,327)(93,326)(94,325)(95,324)(96,323)(97,322)(98,321)(99,320)(100,319)(101,318)(102,317)(103,316)(104,315)(105,314)(106,313)(107,312)(108,311)(109,310)(110,309)(111,308)(112,307)(113,306)(114,305)(115,304)(116,303)(117,302)(118,301)(119,300)(120,299)(121,298)(122,297)(123,296)(124,295)(125,294)(126,293)(127,292)(128,291)(129,290)(130,289)(131,288)(132,287)(133,286)(134,285)(135,284)(136,283)(137,282)(138,281)(139,280)(140,279)(141,278)(142,277)(143,276)(144,275)(145,274)(146,273)(147,272)(148,271)(149,270)(150,269)(151,268)(152,267)(153,266)(154,265)(155,264)(156,263)(157,262)(158,261)(159,260)(160,259)(161,258)(162,257)(163,256)(164,255)(165,254)(166,253)(167,252)(168,251)(169,250)(170,249)(171,248)(172,247)(173,246)(174,245)(175,244)(176,243)(177,242)(178,241)(179,240)(180,239)(181,238)(182,237)(183,236)(184,235)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)(192,227)(193,226)(194,225)(195,224)(196,223)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)Polycyclic presentation:
2,29,359|b^a=b^28,c^a=c^3585207 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20822, 3);