Finite Group DatabasebetaFinite Group Databasebeta| order | 33026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D16513 |
| unambiguous semidirect product | C49⋊D337 |
| unambiguous semidirect product | C337⋊D49 |
| Fitting, derived, Hall | C49×C337 ⋊ C2 |
| Sylow | C337 ⋊ D49 |
| Sylow | C49 ⋊ D337 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C337 | C2 |
| Frattini | C7 | D2359 |
| socle | C7×C337 | D7 |
| LCS | G ▷ C49×C337 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 386 |
| transitive degree | μt(G) = 16513 |
A minimal faithful permutation representation — 2 generators on 386 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);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,386)(52,385)(53,384)(54,383)(55,382)(56,381)(57,380)(58,379)(59,378)(60,377)(61,376)(62,375)(63,374)(64,373)(65,372)(66,371)(67,370)(68,369)(69,368)(70,367)(71,366)(72,365)(73,364)(74,363)(75,362)(76,361)(77,360)(78,359)(79,358)(80,357)(81,356)(82,355)(83,354)(84,353)(85,352)(86,351)(87,350)(88,349)(89,348)(90,347)(91,346)(92,345)(93,344)(94,343)(95,342)(96,341)(97,340)(98,339)(99,338)(100,337)(101,336)(102,335)(103,334)(104,333)(105,332)(106,331)(107,330)(108,329)(109,328)(110,327)(111,326)(112,325)(113,324)(114,323)(115,322)(116,321)(117,320)(118,319)(119,318)(120,317)(121,316)(122,315)(123,314)(124,313)(125,312)(126,311)(127,310)(128,309)(129,308)(130,307)(131,306)(132,305)(133,304)(134,303)(135,302)(136,301)(137,300)(138,299)(139,298)(140,297)(141,296)(142,295)(143,294)(144,293)(145,292)(146,291)(147,290)(148,289)(149,288)(150,287)(151,286)(152,285)(153,284)(154,283)(155,282)(156,281)(157,280)(158,279)(159,278)(160,277)(161,276)(162,275)(163,274)(164,273)(165,272)(166,271)(167,270)(168,269)(169,268)(170,267)(171,266)(172,265)(173,264)(174,263)(175,262)(176,261)(177,260)(178,259)(179,258)(180,257)(181,256)(182,255)(183,254)(184,253)(185,252)(186,251)(187,250)(188,249)(189,248)(190,247)(191,246)(192,245)(193,244)(194,243)(195,242)(196,241)(197,240)(198,239)(199,238)(200,237)(201,236)(202,235)(203,234)(204,233)(205,232)(206,231)(207,230)(208,229)(209,228)(210,227)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)Polycyclic presentation:
2,7,337,7|b^7=d^6,b^a=b^6d,c^a=c^336,d^a=d^68258 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33026, 6);