C₂×D₁₄₀₁₃ #72291364

order	56052
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₈₀₂₆
direct product	C₂×D₁₄₀₁₃

Splittings

Fitting	C₂×C₈₁×C₁₇₃ ⋊ C₂
derived, Hall	C₈₁×C₁₇₃ ⋊ C₂²
Sylow	C₁₇₃ ⋊ C₂×D₈₁
Sylow	C₈₁ ⋊ C₂×D₁₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₀₁₃
commutator, cosocle, hypocenter	C₈₁×C₁₇₃	C₂²
Frattini	C₂₇	C₂×D₅₁₉
Fitting	C₂×C₈₁×C₁₇₃	C₂
socle	C₂×C₃×C₁₇₃	D₂₇
LCS	G ▷ C₈₁×C₁₇₃
UCS	C₁ ◁ C₂

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 256
transitive degree	μ_t(G) = 28026

A minimal faithful permutation representation — 2 generators on 256 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);(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,256)(86,255)(87,254)(88,253)(89,252)(90,251)(91,250)(92,249)(93,248)(94,247)(95,246)(96,245)(97,244)(98,243)(99,242)(100,241)(101,240)(102,239)(103,238)(104,237)(105,236)(106,235)(107,234)(108,233)(109,232)(110,231)(111,230)(112,229)(113,228)(114,227)(115,226)(116,225)(117,224)(118,223)(119,222)(120,221)(121,220)(122,219)(123,218)(124,217)(125,216)(126,215)(127,214)(128,213)(129,212)(130,211)(131,210)(132,209)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,197)(145,196)(146,195)(147,194)(148,193)(149,192)(150,191)(151,190)(152,189)(153,188)(154,187)(155,186)(156,185)(157,184)(158,183)(159,182)(160,181)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171)

Conjugacy classes & character table

14016 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), (4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,256)(86,255)(87,254)(88,253)(89,252)(90,251)(91,250)(92,249)(93,248)(94,247)(95,246)(96,245)(97,244)(98,243)(99,242)(100,241)(101,240)(102,239)(103,238)(104,237)(105,236)(106,235)(107,234)(108,233)(109,232)(110,231)(111,230)(112,229)(113,228)(114,227)(115,226)(116,225)(117,224)(118,223)(119,222)(120,221)(121,220)(122,219)(123,218)(124,217)(125,216)(126,215)(127,214)(128,213)(129,212)(130,211)(131,210)(132,209)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,197)(145,196)(146,195)(147,194)(148,193)(149,192)(150,191)(151,190)(152,189)(153,188)(154,187)(155,186)(156,185)(157,184)(158,183)(159,182)(160,181)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171));

C2×D14013 #72291364