D₁₁₂₄₈ #72215895

order	22496
automorphism	Aut(G) order 58309632
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₂₄₈

Splittings

Fitting	C₁₆×C₁₉×C₃₇ ⋊ C₂
Hall	C₁₉×C₃₇ ⋊ D₁₆
Sylow	C₃₇ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₅₉₂

Subgroups

	subgroup	G/H
center	C₂	D₅₆₂₄
commutator, cosocle	C₈×C₁₉×C₃₇	C₂²
Frattini	C₈	C₂×D₇₀₃
Fitting	C₁₆×C₁₉×C₃₇	C₂
socle	C₂×C₁₉×C₃₇	D₈
hypercenter	C₁₆	D₇₀₃
hypocenter	C₁₉×C₃₇	D₁₆
LCS	G ▷ C₈×C₁₉×C₃₇ ▷ C₄×C₁₉×C₃₇ ▷ C₂×C₁₉×C₃₇ ▷ C₁₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 11248

A minimal faithful permutation representation — 2 generators on 72 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(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:

2^5,19,37|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^36

Conjugacy classes & character table

5627 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₄₉₆
maximal normal subgroup of	D₂₂₄₉₆
maximal quotient of	Dic₁₁₂₄₈, D₂₂₄₉₆

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, 19, 37]);;
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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11248 #72215895