C₂×D₁₅₉₄₃ #72301800

order	63772
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₁₈₈₆
direct product	C₂×D₁₅₉₄₃

Splittings

Fitting, socle	C₂×C₁₀₇×C₁₄₉ ⋊ C₂
derived, Hall	C₁₀₇×C₁₄₉ ⋊ C₂²
Sylow	C₁₄₉ ⋊ C₂×D₁₀₇
Sylow	C₁₀₇ ⋊ C₂×D₁₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₉₄₃
commutator, cosocle, hypocenter	C₁₀₇×C₁₄₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₀₇×C₁₄₉	C₂
LCS	G ▷ C₁₀₇×C₁₄₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 258
transitive degree	μ_t(G) = 31886

A minimal faithful permutation representation — 2 generators on 258 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);(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(111,258)(112,257)(113,256)(114,255)(115,254)(116,253)(117,252)(118,251)(119,250)(120,249)(121,248)(122,247)(123,246)(124,245)(125,244)(126,243)(127,242)(128,241)(129,240)(130,239)(131,238)(132,237)(133,236)(134,235)(135,234)(136,233)(137,232)(138,231)(139,230)(140,229)(141,228)(142,227)(143,226)(144,225)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209)(161,208)(162,207)(163,206)(164,205)(165,204)(166,203)(167,202)(168,201)(169,200)(170,199)(171,198)(172,197)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)

Conjugacy classes & character table

15946 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), (4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(111,258)(112,257)(113,256)(114,255)(115,254)(116,253)(117,252)(118,251)(119,250)(120,249)(121,248)(122,247)(123,246)(124,245)(125,244)(126,243)(127,242)(128,241)(129,240)(130,239)(131,238)(132,237)(133,236)(134,235)(135,234)(136,233)(137,232)(138,231)(139,230)(140,229)(141,228)(142,227)(143,226)(144,225)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209)(161,208)(162,207)(163,206)(164,205)(165,204)(166,203)(167,202)(168,201)(169,200)(170,199)(171,198)(172,197)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185));

C2×D15943 #72301800