Dic₂₁₀₀ #72212445

order	8400
automorphism	Aut(G) order 4032000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₁₀₀

Splittings

Hall	C₃×C₇×C₂₅ ⋊ Q₁₆
Hall	C₇×C₂₅ ⋊ Dic₁₂
Hall	C₃×C₂₅ ⋊ Dic₂₈
Sylow	C₂₅ ⋊ Dic₈₄
Hall	C₃×C₇ ⋊ Dic₁₀₀
Sylow	C₇ ⋊ Dic₃₀₀
Sylow	C₃ ⋊ Dic₇₀₀

Subgroups

	subgroup	G/H
center	C₂	D₂₁₀₀
commutator, cosocle	C₃×C₄×C₇×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₁₀₅
Fitting	C₃×C₇×C₈×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₂₀
hypercenter	C₈	D₅₂₅
hypocenter	C₃×C₇×C₂₅	Q₁₆
LCS	G ▷ C₃×C₄×C₇×C₂₅ ▷ C₂×C₃×C₇×C₂₅ ▷ C₃×C₇×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 8400

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^4,3,5,7,5|a^2=d,b^2=cd,c^2=d,f^5=h,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4h^4,g^a=g^6,h^a=h^4

Conjugacy classes & character table

2103 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₂₀₀, Dic₆₃₀₀, Dic₁₀₅₀₀, …
maximal normal subgroup of	Dic₄₂₀₀
maximal quotient of	Dic₆₃₀₀, Dic₁₀₅₀₀, Dic₁₄₇₀₀

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

Dic2100 #72212445