D₈₉₀₄ #72213764

order	17808
automorphism	Aut(G) order 22224384
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₉₀₄

Splittings

Fitting	C₃×C₇×C₈×C₅₃ ⋊ C₂
Hall	C₃×C₇×C₅₃ ⋊ D₈
Hall	C₇×C₅₃ ⋊ D₂₄
Hall	C₃×C₅₃ ⋊ D₅₆
Sylow	C₅₃ ⋊ D₁₆₈
Hall	C₃×C₇ ⋊ D₄₂₄
Sylow	C₇ ⋊ D₁₂₇₂
Sylow	C₃ ⋊ D₂₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₄₄₅₂
commutator, cosocle	C₃×C₄×C₇×C₅₃	C₂²
Frattini	C₄	C₂×D₁₁₁₃
Fitting	C₃×C₇×C₈×C₅₃	C₂
socle	C₂×C₃×C₇×C₅₃	D₄
hypercenter	C₈	D₁₁₁₃
hypocenter	C₃×C₇×C₅₃	D₈
LCS	G ▷ C₃×C₄×C₇×C₅₃ ▷ C₂×C₃×C₇×C₅₃ ▷ C₃×C₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 8904

A minimal faithful permutation representation — 2 generators on 71 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);(2,8)(3,7)(4,6)(10,11)(13,18)(14,17)(15,16)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^4,3,7,53|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,g^a=g^52

Conjugacy classes & character table

4455 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₈₀₈, D₂₆₇₁₂
maximal normal subgroup of	D₁₇₈₀₈
maximal quotient of	Dic₈₉₀₄, D₁₇₈₀₈, D₂₆₇₁₂

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, 2, 3, 7, 53]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8904 #72213764