D₇₁₀₀ #72212558

order	14200
automorphism	Aut(G) order 19880000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₁₀₀

Splittings

Fitting	C₄×C₂₅×C₇₁ ⋊ C₂
Hall	C₂₅×C₇₁ ⋊ D₄
Sylow	C₇₁ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₂₈₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₇₇₅
commutator, cosocle	C₂×C₂₅×C₇₁	C₂²
Frattini	C₂×C₅	C₂×D₃₅₅
Fitting	C₄×C₂₅×C₇₁	C₂
socle	C₂×C₅×C₇₁	C₂×D₅
hypercenter	C₄	D₁₇₇₅
hypocenter	C₂₅×C₇₁	D₄
LCS	G ▷ C₂×C₂₅×C₇₁ ▷ C₂₅×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 7100

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);(2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Polycyclic presentation:

2^3,5,71,5|b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^70,f^a=f^4

Conjugacy classes & character table

3553 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₄₂₀₀, D₂₁₃₀₀
maximal normal subgroup of	D₁₄₂₀₀
maximal quotient of	D₁₄₂₀₀, Dic₇₁₀₀, D₂₁₃₀₀

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, 5, 71, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7100 #72212558