D₂₁₂₄ #72210992

order	4248
automorphism	Aut(G) order 1478304
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₂₄

Splittings

Fitting	C₄×C₉×C₅₉ ⋊ C₂
Hall	C₉×C₅₉ ⋊ D₄
Sylow	C₅₉ ⋊ D₃₆
Sylow	C₉ ⋊ D₂₃₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₃₁
commutator, cosocle	C₂×C₉×C₅₉	C₂²
Frattini	C₂×C₃	C₂×D₁₇₇
Fitting	C₄×C₉×C₅₉	C₂
socle	C₂×C₃×C₅₉	C₂×D₃
hypercenter	C₄	D₅₃₁
hypocenter	C₉×C₅₉	D₄
LCS	G ▷ C₂×C₉×C₅₉ ▷ C₉×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 2124

A minimal faithful permutation representation — 2 generators on 72 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2^3,3,59,3|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^58,f^a=f^2

Conjugacy classes & character table

1065 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₂₄₈, D₆₃₇₂, D₁₀₆₂₀, …
maximal normal subgroup of	D₄₂₄₈
maximal quotient of	D₄₂₄₈, D₆₃₇₂, Dic₂₁₂₄, …

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, 3, 59, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^58);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2124 #72210992