D₄₆₄₄ #72211468

order	9288
automorphism	Aut(G) order 7021728
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₆₄₄

Splittings

Fitting	C₄×C₂₇×C₄₃ ⋊ C₂
Hall	C₂₇×C₄₃ ⋊ D₄
Sylow	C₄₃ ⋊ D₁₀₈
Sylow	C₂₇ ⋊ D₁₇₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₁₆₁
commutator, cosocle	C₂×C₂₇×C₄₃	C₂²
Frattini	C₂×C₉	C₂×D₁₂₉
Fitting	C₄×C₂₇×C₄₃	C₂
socle	C₂×C₃×C₄₃	C₂×D₉
hypercenter	C₄	D₁₁₆₁
hypocenter	C₂₇×C₄₃	D₄
LCS	G ▷ C₂×C₂₇×C₄₃ ▷ C₂₇×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 4644

A minimal faithful permutation representation — 2 generators on 74 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);(2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^3,3,43,3^2|b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^42,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

2325 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₂₈₈, D₁₃₉₃₂, D₂₃₂₂₀, …
maximal normal subgroup of	D₉₂₈₈
maximal quotient of	D₉₂₈₈, Dic₄₆₄₄, D₁₃₉₃₂, …

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, 3, 43, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4644 #72211468