C₂×D₁₃₅₈₉ #72290504

order	54356
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₇₁₇₈
direct product	C₂×D₁₃₅₈₉

Splittings

Fitting, socle	C₂×C₁₀₇×C₁₂₇ ⋊ C₂
derived, Hall	C₁₀₇×C₁₂₇ ⋊ C₂²
Sylow	C₁₂₇ ⋊ C₂×D₁₀₇
Sylow	C₁₀₇ ⋊ C₂×D₁₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₅₈₉
commutator, cosocle, hypocenter	C₁₀₇×C₁₂₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₀₇×C₁₂₇	C₂
LCS	G ▷ C₁₀₇×C₁₂₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 27178

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);(4,109)(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)(111,236)(112,235)(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)(120,227)(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)

Conjugacy classes & character table

13592 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), (4,109)(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)(111,236)(112,235)(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)(120,227)(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174));

C2×D13589 #72290504