C₂×D₁₀₂₃₃ #72274482

order	40932
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₀₄₆₆
direct product	C₂×D₁₀₂₃₃

Splittings

Fitting	C₂×C₂₇×C₃₇₉ ⋊ C₂
derived, Hall	C₂₇×C₃₇₉ ⋊ C₂²
Sylow	C₃₇₉ ⋊ C₂×D₂₇
Sylow	C₂₇ ⋊ C₂×D₃₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₀₂₃₃
commutator, cosocle, hypocenter	C₂₇×C₃₇₉	C₂²
Frattini	C₉	C₂×D₁₁₃₇
Fitting	C₂×C₂₇×C₃₇₉	C₂
socle	C₂×C₃×C₃₇₉	D₉
LCS	G ▷ C₂₇×C₃₇₉
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 408
transitive degree	μ_t(G) = 20466

A minimal faithful permutation representation — 2 generators on 408 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,408)(32,407)(33,406)(34,405)(35,404)(36,403)(37,402)(38,401)(39,400)(40,399)(41,398)(42,397)(43,396)(44,395)(45,394)(46,393)(47,392)(48,391)(49,390)(50,389)(51,388)(52,387)(53,386)(54,385)(55,384)(56,383)(57,382)(58,381)(59,380)(60,379)(61,378)(62,377)(63,376)(64,375)(65,374)(66,373)(67,372)(68,371)(69,370)(70,369)(71,368)(72,367)(73,366)(74,365)(75,364)(76,363)(77,362)(78,361)(79,360)(80,359)(81,358)(82,357)(83,356)(84,355)(85,354)(86,353)(87,352)(88,351)(89,350)(90,349)(91,348)(92,347)(93,346)(94,345)(95,344)(96,343)(97,342)(98,341)(99,340)(100,339)(101,338)(102,337)(103,336)(104,335)(105,334)(106,333)(107,332)(108,331)(109,330)(110,329)(111,328)(112,327)(113,326)(114,325)(115,324)(116,323)(117,322)(118,321)(119,320)(120,319)(121,318)(122,317)(123,316)(124,315)(125,314)(126,313)(127,312)(128,311)(129,310)(130,309)(131,308)(132,307)(133,306)(134,305)(135,304)(136,303)(137,302)(138,301)(139,300)(140,299)(141,298)(142,297)(143,296)(144,295)(145,294)(146,293)(147,292)(148,291)(149,290)(150,289)(151,288)(152,287)(153,286)(154,285)(155,284)(156,283)(157,282)(158,281)(159,280)(160,279)(161,278)(162,277)(163,276)(164,275)(165,274)(166,273)(167,272)(168,271)(169,270)(170,269)(171,268)(172,267)(173,266)(174,265)(175,264)(176,263)(177,262)(178,261)(179,260)(180,259)(181,258)(182,257)(183,256)(184,255)(185,254)(186,253)(187,252)(188,251)(189,250)(190,249)(191,248)(192,247)(193,246)(194,245)(195,244)(196,243)(197,242)(198,241)(199,240)(200,239)(201,238)(202,237)(203,236)(204,235)(205,234)(206,233)(207,232)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220)

Conjugacy classes & character table

10236 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,408)(32,407)(33,406)(34,405)(35,404)(36,403)(37,402)(38,401)(39,400)(40,399)(41,398)(42,397)(43,396)(44,395)(45,394)(46,393)(47,392)(48,391)(49,390)(50,389)(51,388)(52,387)(53,386)(54,385)(55,384)(56,383)(57,382)(58,381)(59,380)(60,379)(61,378)(62,377)(63,376)(64,375)(65,374)(66,373)(67,372)(68,371)(69,370)(70,369)(71,368)(72,367)(73,366)(74,365)(75,364)(76,363)(77,362)(78,361)(79,360)(80,359)(81,358)(82,357)(83,356)(84,355)(85,354)(86,353)(87,352)(88,351)(89,350)(90,349)(91,348)(92,347)(93,346)(94,345)(95,344)(96,343)(97,342)(98,341)(99,340)(100,339)(101,338)(102,337)(103,336)(104,335)(105,334)(106,333)(107,332)(108,331)(109,330)(110,329)(111,328)(112,327)(113,326)(114,325)(115,324)(116,323)(117,322)(118,321)(119,320)(120,319)(121,318)(122,317)(123,316)(124,315)(125,314)(126,313)(127,312)(128,311)(129,310)(130,309)(131,308)(132,307)(133,306)(134,305)(135,304)(136,303)(137,302)(138,301)(139,300)(140,299)(141,298)(142,297)(143,296)(144,295)(145,294)(146,293)(147,292)(148,291)(149,290)(150,289)(151,288)(152,287)(153,286)(154,285)(155,284)(156,283)(157,282)(158,281)(159,280)(160,279)(161,278)(162,277)(163,276)(164,275)(165,274)(166,273)(167,272)(168,271)(169,270)(170,269)(171,268)(172,267)(173,266)(174,265)(175,264)(176,263)(177,262)(178,261)(179,260)(180,259)(181,258)(182,257)(183,256)(184,255)(185,254)(186,253)(187,252)(188,251)(189,250)(190,249)(191,248)(192,247)(193,246)(194,245)(195,244)(196,243)(197,242)(198,241)(199,240)(200,239)(201,238)(202,237)(203,236)(204,235)(205,234)(206,233)(207,232)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220));

C2×D10233 #72274482