D₁₁₂₈₈ #72215914

order	22576
automorphism	Aut(G) order 59239424
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₂₈₈

Splittings

Fitting	C₈×C₁₇×C₈₃ ⋊ C₂
Hall	C₁₇×C₈₃ ⋊ D₈
Sylow	C₈₃ ⋊ D₁₃₆
Sylow	C₁₇ ⋊ D₆₆₄

Subgroups

	subgroup	G/H
center	C₂	D₅₆₄₄
commutator, cosocle	C₄×C₁₇×C₈₃	C₂²
Frattini	C₄	C₂×D₁₄₁₁
Fitting	C₈×C₁₇×C₈₃	C₂
socle	C₂×C₁₇×C₈₃	D₄
hypercenter	C₈	D₁₄₁₁
hypocenter	C₁₇×C₈₃	D₈
LCS	G ▷ C₄×C₁₇×C₈₃ ▷ C₂×C₁₇×C₈₃ ▷ C₁₇×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 11288

A minimal faithful permutation representation — 2 generators on 108 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108);(2,8)(3,7)(4,6)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^4,17,83|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^16,f^a=f^82

Conjugacy classes & character table

5647 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₅₇₆
maximal normal subgroup of	D₂₂₅₇₆
maximal quotient of	Dic₁₁₂₈₈, D₂₂₅₇₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 17, 83]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11288 #72215914