D₂₆₅₆₄ #72289909

order	53128
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₅₆₄

Splittings

Fitting	C₄×C₂₉×C₂₂₉ ⋊ C₂
Hall	C₂₉×C₂₂₉ ⋊ D₄
Sylow	C₂₂₉ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₉₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₄₁
commutator, socle, cosocle	C₂×C₂₉×C₂₂₉	C₂²
Fitting	C₄×C₂₉×C₂₂₉	C₂
hypercenter	C₄	D₆₆₄₁
hypocenter	C₂₉×C₂₂₉	D₄
LCS	G ▷ C₂×C₂₉×C₂₂₉ ▷ C₂₉×C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 262
transitive degree	μ_t(G) = 26564

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

Polycyclic presentation:

2^3,29,229|b^2=c,b^a=bc,d^a=d^28,e^a=e^228

Conjugacy classes & character table

13285 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 29, 229]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^228);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26564 #72289909