D₃₀₇₄₄ #72296087

order	61488
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₇₄₄

Splittings

Fitting	C₇×C₈×C₉×C₆₁ ⋊ C₂
Hall	C₇×C₉×C₆₁ ⋊ D₈
Hall	C₉×C₆₁ ⋊ D₅₆
Hall	C₇×C₆₁ ⋊ D₇₂
Hall	C₇×C₉ ⋊ D₄₈₈
Sylow	C₆₁ ⋊ D₅₀₄
Sylow	C₉ ⋊ D₃₄₁₆
Sylow	C₇ ⋊ D₄₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₃₇₂
commutator, cosocle	C₄×C₇×C₉×C₆₁	C₂²
Frattini	C₃×C₄	C₂×D₁₂₈₁
Fitting	C₇×C₈×C₉×C₆₁	C₂
socle	C₂×C₃×C₇×C₆₁	D₁₂
hypercenter	C₈	D₃₈₄₃
hypocenter	C₇×C₉×C₆₁	D₈
LCS	G ▷ C₄×C₇×C₉×C₆₁ ▷ C₂×C₇×C₉×C₆₁ ▷ C₇×C₉×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 30744

A minimal faithful permutation representation — 2 generators on 85 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,24)(20,23)(21,22)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^4,3,7,61,3|b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^6,g^a=g^60,h^a=h^2

Conjugacy classes & character table

15375 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 7, 61, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^60);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30744 #72296087