D₈₇₇₂ #72213664

order	17544
automorphism	Aut(G) order 23579136
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₇₂

Splittings

Fitting	C₃×C₄×C₁₇×C₄₃ ⋊ C₂
Hall	C₃×C₁₇×C₄₃ ⋊ D₄
Hall	C₁₇×C₄₃ ⋊ D₁₂
Hall	C₃×C₄₃ ⋊ D₆₈
Hall	C₃×C₁₇ ⋊ D₁₇₂
Sylow	C₄₃ ⋊ D₂₀₄
Sylow	C₁₇ ⋊ D₅₁₆
Sylow	C₃ ⋊ D₂₉₂₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₉₃
commutator, socle, cosocle	C₂×C₃×C₁₇×C₄₃	C₂²
Fitting	C₃×C₄×C₁₇×C₄₃	C₂
hypercenter	C₄	D₂₁₉₃
hypocenter	C₃×C₁₇×C₄₃	D₄
LCS	G ▷ C₂×C₃×C₁₇×C₄₃ ▷ C₃×C₁₇×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 8772

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);(2,4)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2^3,3,17,43|b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^42

Conjugacy classes & character table

4389 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₅₄₄, D₂₆₃₁₆
maximal normal subgroup of	D₁₇₅₄₄
maximal quotient of	Dic₈₇₇₂, D₁₇₅₄₄, D₂₆₃₁₆

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, 17, 43]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8772 #72213664