D₂₇₆₁₂ #72290935

order	55224
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₆₁₂

Splittings

Fitting	C₄×C₉×C₁₃×C₅₉ ⋊ C₂
Hall	C₉×C₁₃×C₅₉ ⋊ D₄
Hall	C₁₃×C₅₉ ⋊ D₃₆
Hall	C₉×C₅₉ ⋊ D₅₂
Hall	C₉×C₁₃ ⋊ D₂₃₆
Sylow	C₅₉ ⋊ D₄₆₈
Sylow	C₁₃ ⋊ D₂₁₂₄
Sylow	C₉ ⋊ D₃₀₆₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₉₀₃
commutator, cosocle	C₂×C₉×C₁₃×C₅₉	C₂²
Frattini	C₂×C₃	C₂×D₂₃₀₁
Fitting	C₄×C₉×C₁₃×C₅₉	C₂
socle	C₂×C₃×C₁₃×C₅₉	C₂×D₃
hypercenter	C₄	D₆₉₀₃
hypocenter	C₉×C₁₃×C₅₉	D₄
LCS	G ▷ C₂×C₉×C₁₃×C₅₉ ▷ C₉×C₁₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 27612

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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,3,13,59,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^12,f^a=f^58,g^a=g^2

Conjugacy classes & character table

13809 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, 3, 13, 59, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27612 #72290935