D₃₀₄₄₄ #72295758

order	60888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₄₄₄

Splittings

Fitting	C₃×C₄×C₄₃×C₅₉ ⋊ C₂
Hall	C₃×C₄₃×C₅₉ ⋊ D₄
Hall	C₄₃×C₅₉ ⋊ D₁₂
Hall	C₃×C₅₉ ⋊ D₁₇₂
Hall	C₃×C₄₃ ⋊ D₂₃₆
Sylow	C₅₉ ⋊ D₅₁₆
Sylow	C₄₃ ⋊ D₇₀₈
Sylow	C₃ ⋊ D₁₀₁₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₆₁₁
commutator, socle, cosocle	C₂×C₃×C₄₃×C₅₉	C₂²
Fitting	C₃×C₄×C₄₃×C₅₉	C₂
hypercenter	C₄	D₇₆₁₁
hypocenter	C₃×C₄₃×C₅₉	D₄
LCS	G ▷ C₂×C₃×C₄₃×C₅₉ ▷ C₃×C₄₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 30444

A minimal faithful permutation representation — 2 generators on 109 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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(2,4)(6,7)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2^3,3,43,59|b^2=c,b^a=bc,d^a=d^2,e^a=e^42,f^a=f^58

Conjugacy classes & character table

15225 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, 3, 43, 59]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30444 #72295758