D₃₀₀₄₄ #72295377

order	60088
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₀₄₄

Splittings

Fitting	C₄×C₇×C₂₉×C₃₇ ⋊ C₂
Hall	C₇×C₂₉×C₃₇ ⋊ D₄
Hall	C₂₉×C₃₇ ⋊ D₂₈
Hall	C₇×C₃₇ ⋊ D₁₁₆
Hall	C₇×C₂₉ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₈₁₂
Sylow	C₂₉ ⋊ D₁₀₃₆
Sylow	C₇ ⋊ D₄₂₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₁₁
commutator, socle, cosocle	C₂×C₇×C₂₉×C₃₇	C₂²
Fitting	C₄×C₇×C₂₉×C₃₇	C₂
hypercenter	C₄	D₇₅₁₁
hypocenter	C₇×C₂₉×C₃₇	D₄
LCS	G ▷ C₂×C₇×C₂₉×C₃₇ ▷ C₇×C₂₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 30044

A minimal faithful permutation representation — 2 generators on 77 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);(2,4)(6,11)(7,10)(8,9)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^3,7,29,37|b^2=c,b^a=bc,d^a=d^6,e^a=e^28,f^a=f^36

Conjugacy classes & character table

15025 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 29, 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.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30044 #72295377