C₂×D₁₅₀₂₃ #72295378

order	60092
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₀₀₄₆
direct product	C₂×D₁₅₀₂₃

Splittings

Fitting, socle	C₂×C₈₃×C₁₈₁ ⋊ C₂
derived, Hall	C₈₃×C₁₈₁ ⋊ C₂²
Sylow	C₁₈₁ ⋊ C₂×D₈₃
Sylow	C₈₃ ⋊ C₂×D₁₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₀₂₃
commutator, cosocle, hypocenter	C₈₃×C₁₈₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₈₃×C₁₈₁	C₂
LCS	G ▷ C₈₃×C₁₈₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 266
transitive degree	μ_t(G) = 30046

A minimal faithful permutation representation — 2 generators on 266 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);(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(87,266)(88,265)(89,264)(90,263)(91,262)(92,261)(93,260)(94,259)(95,258)(96,257)(97,256)(98,255)(99,254)(100,253)(101,252)(102,251)(103,250)(104,249)(105,248)(106,247)(107,246)(108,245)(109,244)(110,243)(111,242)(112,241)(113,240)(114,239)(115,238)(116,237)(117,236)(118,235)(119,234)(120,233)(121,232)(122,231)(123,230)(124,229)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177)

Conjugacy classes & character table

15026 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), (4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(87,266)(88,265)(89,264)(90,263)(91,262)(92,261)(93,260)(94,259)(95,258)(96,257)(97,256)(98,255)(99,254)(100,253)(101,252)(102,251)(103,250)(104,249)(105,248)(106,247)(107,246)(108,245)(109,244)(110,243)(111,242)(112,241)(113,240)(114,239)(115,238)(116,237)(117,236)(118,235)(119,234)(120,233)(121,232)(122,231)(123,230)(124,229)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177));

C2×D15023 #72295378