D₁₁₁₅₂ #72215844

order	22304
automorphism	Aut(G) order 57098240
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₅₂

Splittings

Fitting	C₁₆×C₁₇×C₄₁ ⋊ C₂
Hall	C₁₇×C₄₁ ⋊ D₁₆
Sylow	C₄₁ ⋊ D₂₇₂
Sylow	C₁₇ ⋊ D₆₅₆

Subgroups

	subgroup	G/H
center	C₂	D₅₅₇₆
commutator, cosocle	C₈×C₁₇×C₄₁	C₂²
Frattini	C₈	C₂×D₆₉₇
Fitting	C₁₆×C₁₇×C₄₁	C₂
socle	C₂×C₁₇×C₄₁	D₈
hypercenter	C₁₆	D₆₉₇
hypocenter	C₁₇×C₄₁	D₁₆
LCS	G ▷ C₈×C₁₇×C₄₁ ▷ C₄×C₁₇×C₄₁ ▷ C₂×C₁₇×C₄₁ ▷ C₁₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 11152

A minimal faithful permutation representation — 2 generators on 74 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^5,17,41|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^16,g^a=g^40

Conjugacy classes & character table

5579 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₃₀₄
maximal normal subgroup of	D₂₂₃₀₄
maximal quotient of	Dic₁₁₁₅₂, D₂₂₃₀₄

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, 2, 17, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11152 #72215844