D₁₁₁₇₂ #72215856

order	22344
automorphism	Aut(G) order 33784128
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₇₂

Splittings

Fitting	C₃×C₄×C₁₉×C₄₉ ⋊ C₂
Hall	C₃×C₁₉×C₄₉ ⋊ D₄
Hall	C₁₉×C₄₉ ⋊ D₁₂
Hall	C₃×C₄₉ ⋊ D₇₆
Hall	C₃×C₁₉ ⋊ D₁₉₆
Sylow	C₄₉ ⋊ D₂₂₈
Sylow	C₁₉ ⋊ D₅₈₈
Sylow	C₃ ⋊ D₃₇₂₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₇₉₃
commutator, cosocle	C₂×C₃×C₁₉×C₄₉	C₂²
Frattini	C₂×C₇	C₂×D₃₉₉
Fitting	C₃×C₄×C₁₉×C₄₉	C₂
socle	C₂×C₃×C₇×C₁₉	C₂×D₇
hypercenter	C₄	D₂₇₉₃
hypocenter	C₃×C₁₉×C₄₉	D₄
LCS	G ▷ C₂×C₃×C₁₉×C₄₉ ▷ C₃×C₁₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 11172

A minimal faithful permutation representation — 2 generators on 75 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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,4)(6,7)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^3,3,7,19,7|b^2=c,e^7=g,b^a=bc,d^a=d^2,e^a=e^6g^6,f^a=f^18,g^a=g^6

Conjugacy classes & character table

5589 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₃₄₄
maximal normal subgroup of	D₂₂₃₄₄
maximal quotient of	Dic₁₁₁₇₂, D₂₂₃₄₄

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, 3, 7, 19, 7]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11172 #72215856