Finite Group DatabasebetaFinite Group Databasebeta| order | 33602 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D16801 |
| unambiguous semidirect product | C53⋊D317 |
| unambiguous semidirect product | C317⋊D53 |
| Fitting, socle, derived, Hall | C53×C317 ⋊ C2 |
| Sylow | C317 ⋊ D53 |
| Sylow | C53 ⋊ D317 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C53×C317 | C2 |
| LCS | G ▷ C53×C317 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 370 |
| transitive degree | μt(G) = 16801 |
A minimal faithful permutation representation — 2 generators on 370 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);(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(55,370)(56,369)(57,368)(58,367)(59,366)(60,365)(61,364)(62,363)(63,362)(64,361)(65,360)(66,359)(67,358)(68,357)(69,356)(70,355)(71,354)(72,353)(73,352)(74,351)(75,350)(76,349)(77,348)(78,347)(79,346)(80,345)(81,344)(82,343)(83,342)(84,341)(85,340)(86,339)(87,338)(88,337)(89,336)(90,335)(91,334)(92,333)(93,332)(94,331)(95,330)(96,329)(97,328)(98,327)(99,326)(100,325)(101,324)(102,323)(103,322)(104,321)(105,320)(106,319)(107,318)(108,317)(109,316)(110,315)(111,314)(112,313)(113,312)(114,311)(115,310)(116,309)(117,308)(118,307)(119,306)(120,305)(121,304)(122,303)(123,302)(124,301)(125,300)(126,299)(127,298)(128,297)(129,296)(130,295)(131,294)(132,293)(133,292)(134,291)(135,290)(136,289)(137,288)(138,287)(139,286)(140,285)(141,284)(142,283)(143,282)(144,281)(145,280)(146,279)(147,278)(148,277)(149,276)(150,275)(151,274)(152,273)(153,272)(154,271)(155,270)(156,269)(157,268)(158,267)(159,266)(160,265)(161,264)(162,263)(163,262)(164,261)(165,260)(166,259)(167,258)(168,257)(169,256)(170,255)(171,254)(172,253)(173,252)(174,251)(175,250)(176,249)(177,248)(178,247)(179,246)(180,245)(181,244)(182,243)(183,242)(184,241)(185,240)(186,239)(187,238)(188,237)(189,236)(190,235)(191,234)(192,233)(193,232)(194,231)(195,230)(196,229)(197,228)(198,227)(199,226)(200,225)(201,224)(202,223)(203,222)(204,221)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)Polycyclic presentation:
2,53,317|b^a=b^52,c^a=c^3168402 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33602, 3);