C₂×D₆₇₇₇ #72235520

order	27108
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₃₅₅₄
direct product	C₂×D₆₇₇₇

Splittings

Fitting	C₂×C₂₇×C₂₅₁ ⋊ C₂
derived, Hall	C₂₇×C₂₅₁ ⋊ C₂²
Sylow	C₂₅₁ ⋊ C₂×D₂₇
Sylow	C₂₇ ⋊ C₂×D₂₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₇₇₇
commutator, cosocle, hypocenter	C₂₇×C₂₅₁	C₂²
Frattini	C₉	C₂×D₇₅₃
Fitting	C₂×C₂₇×C₂₅₁	C₂
socle	C₂×C₃×C₂₅₁	D₉
LCS	G ▷ C₂₇×C₂₅₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 280
transitive degree	μ_t(G) = 13554

A minimal faithful permutation representation — 2 generators on 280 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,280)(32,279)(33,278)(34,277)(35,276)(36,275)(37,274)(38,273)(39,272)(40,271)(41,270)(42,269)(43,268)(44,267)(45,266)(46,265)(47,264)(48,263)(49,262)(50,261)(51,260)(52,259)(53,258)(54,257)(55,256)(56,255)(57,254)(58,253)(59,252)(60,251)(61,250)(62,249)(63,248)(64,247)(65,246)(66,245)(67,244)(68,243)(69,242)(70,241)(71,240)(72,239)(73,238)(74,237)(75,236)(76,235)(77,234)(78,233)(79,232)(80,231)(81,230)(82,229)(83,228)(84,227)(85,226)(86,225)(87,224)(88,223)(89,222)(90,221)(91,220)(92,219)(93,218)(94,217)(95,216)(96,215)(97,214)(98,213)(99,212)(100,211)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)(117,194)(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)

Conjugacy classes & character table

6780 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₁₀₈
maximal normal subgroup of	D₂₇₁₀₈
maximal quotient of	Dic₁₃₅₅₄, D₂₇₁₀₈

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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,280)(32,279)(33,278)(34,277)(35,276)(36,275)(37,274)(38,273)(39,272)(40,271)(41,270)(42,269)(43,268)(44,267)(45,266)(46,265)(47,264)(48,263)(49,262)(50,261)(51,260)(52,259)(53,258)(54,257)(55,256)(56,255)(57,254)(58,253)(59,252)(60,251)(61,250)(62,249)(63,248)(64,247)(65,246)(66,245)(67,244)(68,243)(69,242)(70,241)(71,240)(72,239)(73,238)(74,237)(75,236)(76,235)(77,234)(78,233)(79,232)(80,231)(81,230)(82,229)(83,228)(84,227)(85,226)(86,225)(87,224)(88,223)(89,222)(90,221)(91,220)(92,219)(93,218)(94,217)(95,216)(96,215)(97,214)(98,213)(99,212)(100,211)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)(117,194)(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156));

C2×D6777 #72235520