D₂₇₁₀₈ #72290439

order	54216
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₀₈

Splittings

Fitting	C₄×C₂₇×C₂₅₁ ⋊ C₂
Hall	C₂₇×C₂₅₁ ⋊ D₄
Sylow	C₂₅₁ ⋊ D₁₀₈
Sylow	C₂₇ ⋊ D₁₀₀₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₇₇₇
commutator, cosocle	C₂×C₂₇×C₂₅₁	C₂²
Frattini	C₂×C₉	C₂×D₇₅₃
Fitting	C₄×C₂₇×C₂₅₁	C₂
socle	C₂×C₃×C₂₅₁	C₂×D₉
hypercenter	C₄	D₆₇₇₇
hypocenter	C₂₇×C₂₅₁	D₄
LCS	G ▷ C₂×C₂₇×C₂₅₁ ▷ C₂₇×C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 282
transitive degree	μ_t(G) = 27108

A minimal faithful permutation representation — 2 generators on 282 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282);(2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,282)(34,281)(35,280)(36,279)(37,278)(38,277)(39,276)(40,275)(41,274)(42,273)(43,272)(44,271)(45,270)(46,269)(47,268)(48,267)(49,266)(50,265)(51,264)(52,263)(53,262)(54,261)(55,260)(56,259)(57,258)(58,257)(59,256)(60,255)(61,254)(62,253)(63,252)(64,251)(65,250)(66,249)(67,248)(68,247)(69,246)(70,245)(71,244)(72,243)(73,242)(74,241)(75,240)(76,239)(77,238)(78,237)(79,236)(80,235)(81,234)(82,233)(83,232)(84,231)(85,230)(86,229)(87,228)(88,227)(89,226)(90,225)(91,224)(92,223)(93,222)(94,221)(95,220)(96,219)(97,218)(98,217)(99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,210)(106,209)(107,208)(108,207)(109,206)(110,205)(111,204)(112,203)(113,202)(114,201)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,194)(122,193)(123,192)(124,191)(125,190)(126,189)(127,188)(128,187)(129,186)(130,185)(131,184)(132,183)(133,182)(134,181)(135,180)(136,179)(137,178)(138,177)(139,176)(140,175)(141,174)(142,173)(143,172)(144,171)(145,170)(146,169)(147,168)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(157,158)

Polycyclic presentation:

2^3,3,251,3^2|b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^250,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

13557 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 251, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^250);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27108 #72290439