D₂₂₁₀₀ #72275498

order	44200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₁₀₀

Splittings

Fitting	C₄×C₁₃×C₁₇×C₂₅ ⋊ C₂
Hall	C₁₃×C₁₇×C₂₅ ⋊ D₄
Hall	C₁₇×C₂₅ ⋊ D₅₂
Hall	C₁₃×C₂₅ ⋊ D₆₈
Hall	C₁₃×C₁₇ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₈₈₄
Sylow	C₁₇ ⋊ D₁₃₀₀
Sylow	C₁₃ ⋊ D₁₇₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₅₂₅
commutator, cosocle	C₂×C₁₃×C₁₇×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₁₁₀₅
Fitting	C₄×C₁₃×C₁₇×C₂₅	C₂
socle	C₂×C₅×C₁₃×C₁₇	C₂×D₅
hypercenter	C₄	D₅₅₂₅
hypocenter	C₁₃×C₁₇×C₂₅	D₄
LCS	G ▷ C₂×C₁₃×C₁₇×C₂₅ ▷ C₁₃×C₁₇×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 22100

A minimal faithful permutation representation — 2 generators on 59 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);(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)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^3,5,13,17,5|b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^12,f^a=f^16,g^a=g^4

Conjugacy classes & character table

11053 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, 13, 17, 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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D22100 #72275498