D₁₀₃₆ #72210109

order	2072
automorphism	Aut(G) order 447552
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₃₆

Splittings

Fitting	C₄×C₇×C₃₇ ⋊ C₂
Hall	C₇×C₃₇ ⋊ D₄
Sylow	C₃₇ ⋊ D₂₈
Sylow	C₇ ⋊ D₁₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₉
commutator, socle, cosocle	C₂×C₇×C₃₇	C₂²
Fitting	C₄×C₇×C₃₇	C₂
hypercenter	C₄	D₂₅₉
hypocenter	C₇×C₃₇	D₄
LCS	G ▷ C₂×C₇×C₃₇ ▷ C₇×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 1036

A minimal faithful permutation representation — 2 generators on 48 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);(2,4)(6,11)(7,10)(8,9)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Polycyclic presentation:

2^3,7,37|b^2=c,b^a=bc,d^a=d^6,e^a=e^36

Conjugacy classes & character table

521 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₁₀₈, D₂₀₇₂, D₅₁₈₀, …
maximal normal subgroup of	D₂₀₇₂
maximal quotient of	D₃₁₀₈, D₂₀₇₂, D₅₁₈₀, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 37]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1036 #72210109