D₂₆₃₁₂ #72289670

order	52624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₃₁₂

Splittings

Fitting	C₈×C₁₁×C₁₃×C₂₃ ⋊ C₂
Hall	C₁₁×C₁₃×C₂₃ ⋊ D₈
Hall	C₁₃×C₂₃ ⋊ D₈₈
Hall	C₁₁×C₂₃ ⋊ D₁₀₄
Hall	C₁₁×C₁₃ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₁₁₄₄
Sylow	C₁₃ ⋊ D₂₀₂₄
Sylow	C₁₁ ⋊ D₂₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₁₅₆
commutator, cosocle	C₄×C₁₁×C₁₃×C₂₃	C₂²
Frattini	C₄	C₂×D₃₂₈₉
Fitting	C₈×C₁₁×C₁₃×C₂₃	C₂
socle	C₂×C₁₁×C₁₃×C₂₃	D₄
hypercenter	C₈	D₃₂₈₉
hypocenter	C₁₁×C₁₃×C₂₃	D₈
LCS	G ▷ C₄×C₁₁×C₁₃×C₂₃ ▷ C₂×C₁₁×C₁₃×C₂₃ ▷ C₁₁×C₁₃×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 26312

A minimal faithful permutation representation — 2 generators on 55 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);(2,8)(3,7)(4,6)(10,19)(11,18)(12,17)(13,16)(14,15)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^4,11,13,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10,f^a=f^12,g^a=g^22

Conjugacy classes & character table

13159 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, 11, 13, 23]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26312 #72289670