D₁₁₂₂₄ #72215873

order	22448
automorphism	Aut(G) order 59262720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₂₂₄

Splittings

Fitting	C₈×C₂₃×C₆₁ ⋊ C₂
Hall	C₂₃×C₆₁ ⋊ D₈
Sylow	C₆₁ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₅₆₁₂
commutator, cosocle	C₄×C₂₃×C₆₁	C₂²
Frattini	C₄	C₂×D₁₄₀₃
Fitting	C₈×C₂₃×C₆₁	C₂
socle	C₂×C₂₃×C₆₁	D₄
hypercenter	C₈	D₁₄₀₃
hypocenter	C₂₃×C₆₁	D₈
LCS	G ▷ C₄×C₂₃×C₆₁ ▷ C₂×C₂₃×C₆₁ ▷ C₂₃×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 11224

A minimal faithful permutation representation — 2 generators on 92 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);(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)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2^4,23,61|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^60

Conjugacy classes & character table

5615 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₄₄₈
maximal normal subgroup of	D₂₂₄₄₈
maximal quotient of	Dic₁₁₂₂₄, D₂₂₄₄₈

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, 61]);;
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.6^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11224 #72215873