D₁₆₈₇₂ #72272889

order	33744
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₈₇₂

Splittings

Fitting	C₃×C₈×C₁₉×C₃₇ ⋊ C₂
Hall	C₃×C₁₉×C₃₇ ⋊ D₈
Hall	C₁₉×C₃₇ ⋊ D₂₄
Hall	C₃×C₃₇ ⋊ D₁₅₂
Hall	C₃×C₁₉ ⋊ D₂₉₆
Sylow	C₃₇ ⋊ D₄₅₆
Sylow	C₁₉ ⋊ D₈₈₈
Sylow	C₃ ⋊ D₅₆₂₄

Subgroups

	subgroup	G/H
center	C₂	D₈₄₃₆
commutator, cosocle	C₃×C₄×C₁₉×C₃₇	C₂²
Frattini	C₄	C₂×D₂₁₀₉
Fitting	C₃×C₈×C₁₉×C₃₇	C₂
socle	C₂×C₃×C₁₉×C₃₇	D₄
hypercenter	C₈	D₂₁₀₉
hypocenter	C₃×C₁₉×C₃₇	D₈
LCS	G ▷ C₃×C₄×C₁₉×C₃₇ ▷ C₂×C₃×C₁₉×C₃₇ ▷ C₃×C₁₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 16872

A minimal faithful permutation representation — 2 generators on 67 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);(2,8)(3,7)(4,6)(10,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^4,3,19,37|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18,g^a=g^36

Conjugacy classes & character table

8439 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, 3, 19, 37]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D16872 #72272889