D₁₁₁₀₀ #72215825

order	22200
automorphism	Aut(G) order 31968000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₀₀

Splittings

Fitting	C₃×C₄×C₂₅×C₃₇ ⋊ C₂
Hall	C₃×C₂₅×C₃₇ ⋊ D₄
Hall	C₂₅×C₃₇ ⋊ D₁₂
Hall	C₃×C₃₇ ⋊ D₁₀₀
Hall	C₃×C₂₅ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₃₀₀
Sylow	C₂₅ ⋊ D₄₄₄
Sylow	C₃ ⋊ D₃₇₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₇₇₅
commutator, cosocle	C₂×C₃×C₂₅×C₃₇	C₂²
Frattini	C₂×C₅	C₂×D₅₅₅
Fitting	C₃×C₄×C₂₅×C₃₇	C₂
socle	C₂×C₃×C₅×C₃₇	C₂×D₅
hypercenter	C₄	D₂₇₇₅
hypocenter	C₃×C₂₅×C₃₇	D₄
LCS	G ▷ C₂×C₃×C₂₅×C₃₇ ▷ C₃×C₂₅×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 11100

A minimal faithful permutation representation — 2 generators on 69 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);(2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^3,3,5,37,5|b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^36,g^a=g^4

Conjugacy classes & character table

5553 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₀₀
maximal normal subgroup of	D₂₂₂₀₀
maximal quotient of	Dic₁₁₁₀₀, D₂₂₂₀₀

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, 3, 5, 37, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11100 #72215825