D₁₀₄₈₈ #72215403

order	20976
automorphism	Aut(G) order 33225984
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₄₈₈

Splittings

Fitting	C₃×C₈×C₁₉×C₂₃ ⋊ C₂
Hall	C₃×C₁₉×C₂₃ ⋊ D₈
Hall	C₁₉×C₂₃ ⋊ D₂₄
Hall	C₃×C₂₃ ⋊ D₁₅₂
Hall	C₃×C₁₉ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₄₅₆
Sylow	C₁₉ ⋊ D₅₅₂
Sylow	C₃ ⋊ D₃₄₉₆

Subgroups

	subgroup	G/H
center	C₂	D₅₂₄₄
commutator, cosocle	C₃×C₄×C₁₉×C₂₃	C₂²
Frattini	C₄	C₂×D₁₃₁₁
Fitting	C₃×C₈×C₁₉×C₂₃	C₂
socle	C₂×C₃×C₁₉×C₂₃	D₄
hypercenter	C₈	D₁₃₁₁
hypocenter	C₃×C₁₉×C₂₃	D₈
LCS	G ▷ C₃×C₄×C₁₉×C₂₃ ▷ C₂×C₃×C₁₉×C₂₃ ▷ C₃×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 10488

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

Polycyclic presentation:

2^4,3,19,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18,g^a=g^22

Conjugacy classes & character table

5247 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₉₇₆, D₃₁₄₆₄
maximal normal subgroup of	D₂₀₉₇₆
maximal quotient of	Dic₁₀₄₈₈, D₂₀₉₇₆, D₃₁₄₆₄

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, 19, 23]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10488 #72215403