C₂×D₁₅₀₄₁ #72295408

order	60164
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₀₀₈₂
direct product	C₂×D₁₅₀₄₁

Splittings

Fitting	C₂×C₈₉×C₁₆₉ ⋊ C₂
derived, Hall	C₈₉×C₁₆₉ ⋊ C₂²
Sylow	C₁₆₉ ⋊ C₂×D₈₉
Sylow	C₈₉ ⋊ C₂×D₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₅₀₄₁
commutator, cosocle, hypocenter	C₈₉×C₁₆₉	C₂²
Frattini	C₁₃	C₂×D₁₁₅₇
Fitting	C₂×C₈₉×C₁₆₉	C₂
socle	C₂×C₁₃×C₈₉	D₁₃
LCS	G ▷ C₈₉×C₁₆₉
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 260
transitive degree	μ_t(G) = 30082

A minimal faithful permutation representation — 2 generators on 260 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);(4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,160)(16,159)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(173,260)(174,259)(175,258)(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)

Conjugacy classes & character table

15044 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), (4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,160)(16,159)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(173,260)(174,259)(175,258)(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217));

C2×D15041 #72295408