D₂₇₁₀₀ #72290432

order	54200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₀₀

Splittings

Fitting	C₄×C₂₅×C₂₇₁ ⋊ C₂
Hall	C₂₅×C₂₇₁ ⋊ D₄
Sylow	C₂₇₁ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₁₀₈₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₇₇₅
commutator, cosocle	C₂×C₂₅×C₂₇₁	C₂²
Frattini	C₂×C₅	C₂×D₁₃₅₅
Fitting	C₄×C₂₅×C₂₇₁	C₂
socle	C₂×C₅×C₂₇₁	C₂×D₅
hypercenter	C₄	D₆₇₇₅
hypocenter	C₂₅×C₂₇₁	D₄
LCS	G ▷ C₂×C₂₅×C₂₇₁ ▷ C₂₅×C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 300
transitive degree	μ_t(G) = 27100

A minimal faithful permutation representation — 2 generators on 300 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300);(2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,300)(32,299)(33,298)(34,297)(35,296)(36,295)(37,294)(38,293)(39,292)(40,291)(41,290)(42,289)(43,288)(44,287)(45,286)(46,285)(47,284)(48,283)(49,282)(50,281)(51,280)(52,279)(53,278)(54,277)(55,276)(56,275)(57,274)(58,273)(59,272)(60,271)(61,270)(62,269)(63,268)(64,267)(65,266)(66,265)(67,264)(68,263)(69,262)(70,261)(71,260)(72,259)(73,258)(74,257)(75,256)(76,255)(77,254)(78,253)(79,252)(80,251)(81,250)(82,249)(83,248)(84,247)(85,246)(86,245)(87,244)(88,243)(89,242)(90,241)(91,240)(92,239)(93,238)(94,237)(95,236)(96,235)(97,234)(98,233)(99,232)(100,231)(101,230)(102,229)(103,228)(104,227)(105,226)(106,225)(107,224)(108,223)(109,222)(110,221)(111,220)(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)(119,212)(120,211)(121,210)(122,209)(123,208)(124,207)(125,206)(126,205)(127,204)(128,203)(129,202)(130,201)(131,200)(132,199)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)(141,190)(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)

Polycyclic presentation:

2^3,5,271,5|b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^270,f^a=f^4

Conjugacy classes & character table

13553 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 271, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^270);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27100 #72290432