D₂₆₈₀₀ #72290142

order	53600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₈₀₀

Splittings

Fitting	C₁₆×C₂₅×C₆₇ ⋊ C₂
Hall	C₂₅×C₆₇ ⋊ D₁₆
Sylow	C₆₇ ⋊ D₄₀₀
Sylow	C₂₅ ⋊ D₁₀₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₄₀₀
commutator, cosocle	C₈×C₂₅×C₆₇	C₂²
Frattini	C₅×C₈	C₂×D₃₃₅
Fitting	C₁₆×C₂₅×C₆₇	C₂
socle	C₂×C₅×C₆₇	D₄₀
hypercenter	C₁₆	D₁₆₇₅
hypocenter	C₂₅×C₆₇	D₁₆
LCS	G ▷ C₈×C₂₅×C₆₇ ▷ C₄×C₂₅×C₆₇ ▷ C₂×C₂₅×C₆₇ ▷ C₂₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 26800

A minimal faithful permutation representation — 2 generators on 108 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^5,5,67,5|b^2=cd,c^2=de,d^2=e,f^5=h,b^a=bc,c^a=cd,d^a=de,f^a=f^4h^4,g^a=g^66,h^a=h^4

Conjugacy classes & character table

13403 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 67, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
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, 6, 1, F.6^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^66);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26800 #72290142