D₂₇₁₀₄ #72290437

order	54208
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₀₄

Splittings

Fitting	C₇×C₃₂×C₁₂₁ ⋊ C₂
Hall	C₇×C₁₂₁ ⋊ D₃₂
Sylow	C₁₂₁ ⋊ D₂₂₄
Sylow	C₇ ⋊ D₃₈₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₅₅₂
commutator, cosocle	C₇×C₁₆×C₁₂₁	C₂²
Frattini	C₁₁×C₁₆	C₂×D₇₇
Fitting	C₇×C₃₂×C₁₂₁	C₂
socle	C₂×C₇×C₁₁	D₁₇₆
hypercenter	C₃₂	D₈₄₇
hypocenter	C₇×C₁₂₁	D₃₂
LCS	G ▷ C₇×C₁₆×C₁₂₁ ▷ C₇×C₈×C₁₂₁ ▷ C₄×C₇×C₁₂₁ ▷ C₂×C₇×C₁₂₁ ▷ C₇×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 27104

A minimal faithful permutation representation — 2 generators on 160 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,39)(35,38)(36,37)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)

Polycyclic presentation:

2^6,7,11^2|b^2=cd,c^2=de,d^2=ef,e^2=f,h^11=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6,h^a=h^10i^10,i^a=i^10

Conjugacy classes & character table

13555 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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 7, 11, 11]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^10*F.9^10);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27104 #72290437