D₁₁₈₀₀ #72223967

order	23600
automorphism	Aut(G) order 54752000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₀₀

Splittings

Fitting	C₈×C₂₅×C₅₉ ⋊ C₂
Hall	C₂₅×C₅₉ ⋊ D₈
Sylow	C₅₉ ⋊ D₂₀₀
Sylow	C₂₅ ⋊ D₄₇₂

Subgroups

	subgroup	G/H
center	C₂	D₅₉₀₀
commutator, cosocle	C₄×C₂₅×C₅₉	C₂²
Frattini	C₄×C₅	C₂×D₂₉₅
Fitting	C₈×C₂₅×C₅₉	C₂
socle	C₂×C₅×C₅₉	D₂₀
hypercenter	C₈	D₁₄₇₅
hypocenter	C₂₅×C₅₉	D₈
LCS	G ▷ C₄×C₂₅×C₅₉ ▷ C₂×C₂₅×C₅₉ ▷ C₂₅×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 11800

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

Polycyclic presentation:

2^4,5,59,5|b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^58,g^a=g^4

Conjugacy classes & character table

5903 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₀₀
maximal normal subgroup of	D₂₃₆₀₀
maximal quotient of	Dic₁₁₈₀₀, D₂₃₆₀₀

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, 5, 59, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11800 #72223967