C₂×D₁₃₆₇₃ #72290656

order	54692
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₇₃₄₆
direct product	C₂×D₁₃₆₇₃

Splittings

Fitting	C₂×C₁₁₃×C₁₂₁ ⋊ C₂
derived, Hall	C₁₁₃×C₁₂₁ ⋊ C₂²
Sylow	C₁₂₁ ⋊ C₂×D₁₁₃
Sylow	C₁₁₃ ⋊ C₂×D₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₆₇₃
commutator, cosocle, hypocenter	C₁₁₃×C₁₂₁	C₂²
Frattini	C₁₁	C₂×D₁₂₄₃
Fitting	C₂×C₁₁₃×C₁₂₁	C₂
socle	C₂×C₁₁×C₁₁₃	D₁₁
LCS	G ▷ C₁₁₃×C₁₂₁
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 27346

A minimal faithful permutation representation — 2 generators on 236 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);(4,123)(5,122)(6,121)(7,120)(8,119)(9,118)(10,117)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(125,236)(126,235)(127,234)(128,233)(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)(161,200)(162,199)(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)

Conjugacy classes & character table

13676 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), (4,123)(5,122)(6,121)(7,120)(8,119)(9,118)(10,117)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(125,236)(126,235)(127,234)(128,233)(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)(161,200)(162,199)(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181));

C2×D13673 #72290656