D₁₃₀₆₄ #72235342

order	26128
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₀₆₄

Splittings

Fitting	C₈×C₂₃×C₇₁ ⋊ C₂
Hall	C₂₃×C₇₁ ⋊ D₈
Sylow	C₇₁ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₅₆₈

Subgroups

	subgroup	G/H
center	C₂	D₆₅₃₂
commutator, cosocle	C₄×C₂₃×C₇₁	C₂²
Frattini	C₄	C₂×D₁₆₃₃
Fitting	C₈×C₂₃×C₇₁	C₂
socle	C₂×C₂₃×C₇₁	D₄
hypercenter	C₈	D₁₆₃₃
hypocenter	C₂₃×C₇₁	D₈
LCS	G ▷ C₄×C₂₃×C₇₁ ▷ C₂×C₂₃×C₇₁ ▷ C₂₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 13064

A minimal faithful permutation representation — 2 generators on 102 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);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^4,23,71|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^70

Conjugacy classes & character table

6535 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₆₁₂₈
maximal normal subgroup of	D₂₆₁₂₈
maximal quotient of	Dic₁₃₀₆₄, D₂₆₁₂₈

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, 2, 23, 71]);;
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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13064 #72235342