D₂₇₄₁₂ #72290720

order	54824
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₄₁₂

Splittings

Fitting	C₄×C₇×C₁₁×C₈₉ ⋊ C₂
Hall	C₇×C₁₁×C₈₉ ⋊ D₄
Hall	C₁₁×C₈₉ ⋊ D₂₈
Hall	C₇×C₈₉ ⋊ D₄₄
Sylow	C₈₉ ⋊ D₃₀₈
Hall	C₇×C₁₁ ⋊ D₃₅₆
Sylow	C₁₁ ⋊ D₂₄₉₂
Sylow	C₇ ⋊ D₃₉₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₅₃
commutator, socle, cosocle	C₂×C₇×C₁₁×C₈₉	C₂²
Fitting	C₄×C₇×C₁₁×C₈₉	C₂
hypercenter	C₄	D₆₈₅₃
hypocenter	C₇×C₁₁×C₈₉	D₄
LCS	G ▷ C₂×C₇×C₁₁×C₈₉ ▷ C₇×C₁₁×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 27412

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2,3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^3,7,11,89|b^2=c,b^a=bc,d^a=d^6,e^a=e^10,f^a=f^88

Conjugacy classes & character table

13709 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, 11, 89]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27412 #72290720