D₁₁₀₉₆ #72215822

order	22192
automorphism	Aut(G) order 57521664
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₀₉₆

Splittings

Fitting	C₈×C₁₉×C₇₃ ⋊ C₂
Hall	C₁₉×C₇₃ ⋊ D₈
Sylow	C₇₃ ⋊ D₁₅₂
Sylow	C₁₉ ⋊ D₅₈₄

Subgroups

	subgroup	G/H
center	C₂	D₅₅₄₈
commutator, cosocle	C₄×C₁₉×C₇₃	C₂²
Frattini	C₄	C₂×D₁₃₈₇
Fitting	C₈×C₁₉×C₇₃	C₂
socle	C₂×C₁₉×C₇₃	D₄
hypercenter	C₈	D₁₃₈₇
hypocenter	C₁₉×C₇₃	D₈
LCS	G ▷ C₄×C₁₉×C₇₃ ▷ C₂×C₁₉×C₇₃ ▷ C₁₉×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 11096

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

Polycyclic presentation:

2^4,19,73|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^18,f^a=f^72

Conjugacy classes & character table

5551 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₁₉₂
maximal normal subgroup of	D₂₂₁₉₂
maximal quotient of	Dic₁₁₀₉₆, D₂₂₁₉₂

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, 19, 73]);;
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.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11096 #72215822