C₂×D₁₆₃₀₉ #72303702

order	65236
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₂₆₁₈
direct product	C₂×D₁₆₃₀₉

Splittings

Fitting, socle	C₂×C₄₇×C₃₄₇ ⋊ C₂
derived, Hall	C₄₇×C₃₄₇ ⋊ C₂²
Sylow	C₃₄₇ ⋊ C₂×D₄₇
Sylow	C₄₇ ⋊ C₂×D₃₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₃₀₉
commutator, cosocle, hypocenter	C₄₇×C₃₄₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₄₇×C₃₄₇	C₂
LCS	G ▷ C₄₇×C₃₄₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 396
transitive degree	μ_t(G) = 32618

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

Conjugacy classes & character table

16312 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×D16309 #72303702