D₂₇₀₀ #72210609

order	5400
automorphism	Aut(G) order 1944000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₀₀

Splittings

Fitting	C₄×C₂₅×C₂₇ ⋊ C₂
Hall	C₂₅×C₂₇ ⋊ D₄
Sylow	C₂₇ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₁₀₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₇₅
commutator, cosocle	C₂×C₂₅×C₂₇	C₂²
Frattini	C₂×C₅×C₉	C₂×D₁₅
Fitting	C₄×C₂₅×C₂₇	C₂
socle	C₂×C₃×C₅	C₂×D₄₅
hypercenter	C₄	D₆₇₅
hypocenter	C₂₅×C₂₇	D₄
LCS	G ▷ C₂×C₂₅×C₂₇ ▷ C₂₅×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 2700

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);(2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^3,3,5,3,5,3|b^2=c,d^3=f,e^5=g,f^3=h,b^a=bc,d^a=d^2f^2h^2,e^a=e^4g^4,f^a=f^2h^2,g^a=g^4,h^a=h^2

Conjugacy classes & character table

1353 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₄₀₀, D₈₁₀₀, D₁₃₅₀₀, …
maximal normal subgroup of	D₅₄₀₀
maximal quotient of	D₅₄₀₀, Dic₂₇₀₀, D₈₁₀₀, …

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 3, 5, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2700 #72210609