D₂₆₄₈₈ #72289840

order	52976
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₄₈₈

Splittings

Fitting	C₇×C₈×C₁₁×C₄₃ ⋊ C₂
Hall	C₇×C₁₁×C₄₃ ⋊ D₈
Hall	C₁₁×C₄₃ ⋊ D₅₆
Hall	C₇×C₄₃ ⋊ D₈₈
Hall	C₇×C₁₁ ⋊ D₃₄₄
Sylow	C₄₃ ⋊ D₆₁₆
Sylow	C₁₁ ⋊ D₂₄₀₈
Sylow	C₇ ⋊ D₃₇₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₂₄₄
commutator, cosocle	C₄×C₇×C₁₁×C₄₃	C₂²
Frattini	C₄	C₂×D₃₃₁₁
Fitting	C₇×C₈×C₁₁×C₄₃	C₂
socle	C₂×C₇×C₁₁×C₄₃	D₄
hypercenter	C₈	D₃₃₁₁
hypocenter	C₇×C₁₁×C₄₃	D₈
LCS	G ▷ C₄×C₇×C₁₁×C₄₃ ▷ C₂×C₇×C₁₁×C₄₃ ▷ C₇×C₁₁×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 26488

A minimal faithful permutation representation — 2 generators on 69 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,26)(18,25)(19,24)(20,23)(21,22)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^4,7,11,43|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^10,g^a=g^42

Conjugacy classes & character table

13247 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, 7, 11, 43]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26488 #72289840