D₁₃₇₇₆ #72235612

order	27552
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₇₇₆

Splittings

Fitting	C₃×C₇×C₁₆×C₄₁ ⋊ C₂
Hall	C₃×C₇×C₄₁ ⋊ D₁₆
Hall	C₇×C₄₁ ⋊ D₄₈
Hall	C₃×C₄₁ ⋊ D₁₁₂
Sylow	C₄₁ ⋊ D₃₃₆
Hall	C₃×C₇ ⋊ D₆₅₆
Sylow	C₇ ⋊ D₁₉₆₈
Sylow	C₃ ⋊ D₄₅₉₂

Subgroups

	subgroup	G/H
center	C₂	D₆₈₈₈
commutator, cosocle	C₃×C₇×C₈×C₄₁	C₂²
Frattini	C₈	C₂×D₈₆₁
Fitting	C₃×C₇×C₁₆×C₄₁	C₂
socle	C₂×C₃×C₇×C₄₁	D₈
hypercenter	C₁₆	D₈₆₁
hypocenter	C₃×C₇×C₄₁	D₁₆
LCS	G ▷ C₃×C₇×C₈×C₄₁ ▷ C₃×C₄×C₇×C₄₁ ▷ C₂×C₃×C₇×C₄₁ ▷ C₃×C₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 13776

A minimal faithful permutation representation — 2 generators on 67 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,26)(22,25)(23,24)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^5,3,7,41|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^6,h^a=h^40

Conjugacy classes & character table

6891 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₅₅₂
maximal normal subgroup of	D₂₇₅₅₂
maximal quotient of	Dic₁₃₇₇₆, D₂₇₅₅₂

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, 2, 3, 7, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13776 #72235612