C₂×D₁₃₉₆₉ #72291252

order	55876
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₇₉₃₈
direct product	C₂×D₁₃₉₆₉

Splittings

Fitting, socle	C₂×C₆₁×C₂₂₉ ⋊ C₂
derived, Hall	C₆₁×C₂₂₉ ⋊ C₂²
Sylow	C₂₂₉ ⋊ C₂×D₆₁
Sylow	C₆₁ ⋊ C₂×D₂₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₉₆₉
commutator, cosocle, hypocenter	C₆₁×C₂₂₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₆₁×C₂₂₉	C₂
LCS	G ▷ C₆₁×C₂₂₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 292
transitive degree	μ_t(G) = 27938

A minimal faithful permutation representation — 2 generators on 292 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);(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(65,292)(66,291)(67,290)(68,289)(69,288)(70,287)(71,286)(72,285)(73,284)(74,283)(75,282)(76,281)(77,280)(78,279)(79,278)(80,277)(81,276)(82,275)(83,274)(84,273)(85,272)(86,271)(87,270)(88,269)(89,268)(90,267)(91,266)(92,265)(93,264)(94,263)(95,262)(96,261)(97,260)(98,259)(99,258)(100,257)(101,256)(102,255)(103,254)(104,253)(105,252)(106,251)(107,250)(108,249)(109,248)(110,247)(111,246)(112,245)(113,244)(114,243)(115,242)(116,241)(117,240)(118,239)(119,238)(120,237)(121,236)(122,235)(123,234)(124,233)(125,232)(126,231)(127,230)(128,229)(129,228)(130,227)(131,226)(132,225)(133,224)(134,223)(135,222)(136,221)(137,220)(138,219)(139,218)(140,217)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,210)(148,209)(149,208)(150,207)(151,206)(152,205)(153,204)(154,203)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,196)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189)(169,188)(170,187)(171,186)(172,185)(173,184)(174,183)(175,182)(176,181)(177,180)(178,179)

Conjugacy classes & character table

13972 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), (4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(65,292)(66,291)(67,290)(68,289)(69,288)(70,287)(71,286)(72,285)(73,284)(74,283)(75,282)(76,281)(77,280)(78,279)(79,278)(80,277)(81,276)(82,275)(83,274)(84,273)(85,272)(86,271)(87,270)(88,269)(89,268)(90,267)(91,266)(92,265)(93,264)(94,263)(95,262)(96,261)(97,260)(98,259)(99,258)(100,257)(101,256)(102,255)(103,254)(104,253)(105,252)(106,251)(107,250)(108,249)(109,248)(110,247)(111,246)(112,245)(113,244)(114,243)(115,242)(116,241)(117,240)(118,239)(119,238)(120,237)(121,236)(122,235)(123,234)(124,233)(125,232)(126,231)(127,230)(128,229)(129,228)(130,227)(131,226)(132,225)(133,224)(134,223)(135,222)(136,221)(137,220)(138,219)(139,218)(140,217)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,210)(148,209)(149,208)(150,207)(151,206)(152,205)(153,204)(154,203)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,196)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189)(169,188)(170,187)(171,186)(172,185)(173,184)(174,183)(175,182)(176,181)(177,180)(178,179));

C2×D13969 #72291252