D₁₁₇₆ #72210129

order	2352
automorphism	Aut(G) order 395136
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₆

Splittings

Fitting	C₃×C₈×C₄₉ ⋊ C₂
Hall	C₃×C₄₉ ⋊ D₈
Sylow	C₄₉ ⋊ D₂₄
Sylow	C₃ ⋊ D₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₅₈₈
commutator, cosocle	C₃×C₄×C₄₉	C₂²
Frattini	C₄×C₇	C₂×D₂₁
Fitting	C₃×C₈×C₄₉	C₂
socle	C₂×C₃×C₇	D₂₈
hypercenter	C₈	D₁₄₇
hypocenter	C₃×C₄₉	D₈
LCS	G ▷ C₃×C₄×C₄₉ ▷ C₂×C₃×C₄₉ ▷ C₃×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 1176

A minimal faithful permutation representation — 2 generators on 60 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);(2,8)(3,7)(4,6)(10,11)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

591 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₃₅₂, D₃₅₂₈, D₅₈₈₀, …
maximal normal subgroup of	D₂₃₅₂
maximal quotient of	D₂₃₅₂, D₃₅₂₈, Dic₁₁₇₆, …

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

D1176 #72210129