C₂×D₁₆₂₇₉ #72303142

order	65116
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₂₅₅₈
direct product	C₂×D₁₆₂₇₉

Splittings

Fitting, socle	C₂×C₇₃×C₂₂₃ ⋊ C₂
derived, Hall	C₇₃×C₂₂₃ ⋊ C₂²
Sylow	C₂₂₃ ⋊ C₂×D₇₃
Sylow	C₇₃ ⋊ C₂×D₂₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₂₇₉
commutator, cosocle, hypocenter	C₇₃×C₂₂₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇₃×C₂₂₃	C₂
LCS	G ▷ C₇₃×C₂₂₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 298
transitive degree	μ_t(G) = 32558

A minimal faithful permutation representation — 2 generators on 298 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);(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(77,298)(78,297)(79,296)(80,295)(81,294)(82,293)(83,292)(84,291)(85,290)(86,289)(87,288)(88,287)(89,286)(90,285)(91,284)(92,283)(93,282)(94,281)(95,280)(96,279)(97,278)(98,277)(99,276)(100,275)(101,274)(102,273)(103,272)(104,271)(105,270)(106,269)(107,268)(108,267)(109,266)(110,265)(111,264)(112,263)(113,262)(114,261)(115,260)(116,259)(117,258)(118,257)(119,256)(120,255)(121,254)(122,253)(123,252)(124,251)(125,250)(126,249)(127,248)(128,247)(129,246)(130,245)(131,244)(132,243)(133,242)(134,241)(135,240)(136,239)(137,238)(138,237)(139,236)(140,235)(141,234)(142,233)(143,232)(144,231)(145,230)(146,229)(147,228)(148,227)(149,226)(150,225)(151,224)(152,223)(153,222)(154,221)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)

Conjugacy classes & character table

16282 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), (4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(77,298)(78,297)(79,296)(80,295)(81,294)(82,293)(83,292)(84,291)(85,290)(86,289)(87,288)(88,287)(89,286)(90,285)(91,284)(92,283)(93,282)(94,281)(95,280)(96,279)(97,278)(98,277)(99,276)(100,275)(101,274)(102,273)(103,272)(104,271)(105,270)(106,269)(107,268)(108,267)(109,266)(110,265)(111,264)(112,263)(113,262)(114,261)(115,260)(116,259)(117,258)(118,257)(119,256)(120,255)(121,254)(122,253)(123,252)(124,251)(125,250)(126,249)(127,248)(128,247)(129,246)(130,245)(131,244)(132,243)(133,242)(134,241)(135,240)(136,239)(137,238)(138,237)(139,236)(140,235)(141,234)(142,233)(143,232)(144,231)(145,230)(146,229)(147,228)(148,227)(149,226)(150,225)(151,224)(152,223)(153,222)(154,221)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188));

C2×D16279 #72303142