D₂₆₇₆₀ #72290107

order	53520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₇₆₀

Splittings

Fitting	C₃×C₅×C₈×C₂₂₃ ⋊ C₂
Hall	C₃×C₅×C₂₂₃ ⋊ D₈
Hall	C₅×C₂₂₃ ⋊ D₂₄
Hall	C₃×C₂₂₃ ⋊ D₄₀
Sylow	C₂₂₃ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₁₇₈₄
Sylow	C₅ ⋊ D₅₃₅₂
Sylow	C₃ ⋊ D₈₉₂₀

Subgroups

	subgroup	G/H
center	C₂	D₁₃₃₈₀
commutator, cosocle	C₃×C₄×C₅×C₂₂₃	C₂²
Frattini	C₄	C₂×D₃₃₄₅
Fitting	C₃×C₅×C₈×C₂₂₃	C₂
socle	C₂×C₃×C₅×C₂₂₃	D₄
hypercenter	C₈	D₃₃₄₅
hypocenter	C₃×C₅×C₂₂₃	D₈
LCS	G ▷ C₃×C₄×C₅×C₂₂₃ ▷ C₂×C₃×C₅×C₂₂₃ ▷ C₃×C₅×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 239
transitive degree	μ_t(G) = 26760

A minimal faithful permutation representation — 2 generators on 239 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,239)(19,238)(20,237)(21,236)(22,235)(23,234)(24,233)(25,232)(26,231)(27,230)(28,229)(29,228)(30,227)(31,226)(32,225)(33,224)(34,223)(35,222)(36,221)(37,220)(38,219)(39,218)(40,217)(41,216)(42,215)(43,214)(44,213)(45,212)(46,211)(47,210)(48,209)(49,208)(50,207)(51,206)(52,205)(53,204)(54,203)(55,202)(56,201)(57,200)(58,199)(59,198)(60,197)(61,196)(62,195)(63,194)(64,193)(65,192)(66,191)(67,190)(68,189)(69,188)(70,187)(71,186)(72,185)(73,184)(74,183)(75,182)(76,181)(77,180)(78,179)(79,178)(80,177)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)

Polycyclic presentation:

2^4,3,5,223|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^222

Conjugacy classes & character table

13383 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, 2, 3, 5, 223]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26760 #72290107