D₃₀₁₀₀ #72295421

order	60200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₀₀

Splittings

Fitting	C₄×C₇×C₂₅×C₄₃ ⋊ C₂
Hall	C₇×C₂₅×C₄₃ ⋊ D₄
Hall	C₂₅×C₄₃ ⋊ D₂₈
Hall	C₇×C₄₃ ⋊ D₁₀₀
Hall	C₇×C₂₅ ⋊ D₁₇₂
Sylow	C₄₃ ⋊ D₇₀₀
Sylow	C₂₅ ⋊ D₁₂₀₄
Sylow	C₇ ⋊ D₄₃₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₅₂₅
commutator, cosocle	C₂×C₇×C₂₅×C₄₃	C₂²
Frattini	C₂×C₅	C₂×D₁₅₀₅
Fitting	C₄×C₇×C₂₅×C₄₃	C₂
socle	C₂×C₅×C₇×C₄₃	C₂×D₅
hypercenter	C₄	D₇₅₂₅
hypocenter	C₇×C₂₅×C₄₃	D₄
LCS	G ▷ C₂×C₇×C₂₅×C₄₃ ▷ C₇×C₂₅×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 30100

A minimal faithful permutation representation — 2 generators on 79 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);(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)(31,36)(32,35)(33,34)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^3,5,7,43,5|b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^6,f^a=f^42,g^a=g^4

Conjugacy classes & character table

15053 conjugacy classes. The character table is too large to display.

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, 5, 7, 43, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30100 #72295421