D₄₃₁₂ #72211368

order	8624
automorphism	Aut(G) order 7244160
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₃₁₂

Splittings

Fitting	C₈×C₁₁×C₄₉ ⋊ C₂
Hall	C₁₁×C₄₉ ⋊ D₈
Sylow	C₄₉ ⋊ D₈₈
Sylow	C₁₁ ⋊ D₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₂₁₅₆
commutator, cosocle	C₄×C₁₁×C₄₉	C₂²
Frattini	C₄×C₇	C₂×D₇₇
Fitting	C₈×C₁₁×C₄₉	C₂
socle	C₂×C₇×C₁₁	D₂₈
hypercenter	C₈	D₅₃₉
hypocenter	C₁₁×C₄₉	D₈
LCS	G ▷ C₄×C₁₁×C₄₉ ▷ C₂×C₁₁×C₄₉ ▷ C₁₁×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 4312

A minimal faithful permutation representation — 2 generators on 68 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);(2,8)(3,7)(4,6)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^4,7,11,7|b^2=cd,c^2=d,e^7=g,b^a=bc,c^a=cd,e^a=e^6g^6,f^a=f^10,g^a=g^6

Conjugacy classes & character table

2159 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₆₂₄, D₁₂₉₃₆, D₂₁₅₆₀, …
maximal normal subgroup of	D₈₆₂₄
maximal quotient of	D₈₆₂₄, Dic₄₃₁₂, D₁₂₉₃₆, …

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, 7, 11, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, 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.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4312 #72211368