D₄₆₀₀ #72211451

order	9200
automorphism	Aut(G) order 8096000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₆₀₀

Splittings

Fitting	C₈×C₂₃×C₂₅ ⋊ C₂
Hall	C₂₃×C₂₅ ⋊ D₈
Sylow	C₂₅ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₂₀₀

Subgroups

	subgroup	G/H
center	C₂	D₂₃₀₀
commutator, cosocle	C₄×C₂₃×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₁₁₅
Fitting	C₈×C₂₃×C₂₅	C₂
socle	C₂×C₅×C₂₃	D₂₀
hypercenter	C₈	D₅₇₅
hypocenter	C₂₃×C₂₅	D₈
LCS	G ▷ C₄×C₂₃×C₂₅ ▷ C₂×C₂₃×C₂₅ ▷ C₂₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 4600

A minimal faithful permutation representation — 2 generators on 56 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);(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)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^4,5,23,5|b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^22,g^a=g^4

Conjugacy classes & character table

2303 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₂₀₀, D₁₃₈₀₀, D₂₃₀₀₀, …
maximal normal subgroup of	D₉₂₀₀
maximal quotient of	D₉₂₀₀, Dic₄₆₀₀, D₁₃₈₀₀, …

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, 23, 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.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4600 #72211451