D₂₂₆₄₄ #72277408

order	45288
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₆₄₄

Splittings

Fitting	C₄×C₉×C₁₇×C₃₇ ⋊ C₂
Hall	C₉×C₁₇×C₃₇ ⋊ D₄
Hall	C₁₇×C₃₇ ⋊ D₃₆
Hall	C₉×C₃₇ ⋊ D₆₈
Hall	C₉×C₁₇ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₆₁₂
Sylow	C₁₇ ⋊ D₁₃₃₂
Sylow	C₉ ⋊ D₂₅₁₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₆₆₁
commutator, cosocle	C₂×C₉×C₁₇×C₃₇	C₂²
Frattini	C₂×C₃	C₂×D₁₈₈₇
Fitting	C₄×C₉×C₁₇×C₃₇	C₂
socle	C₂×C₃×C₁₇×C₃₇	C₂×D₃
hypercenter	C₄	D₅₆₆₁
hypocenter	C₉×C₁₇×C₃₇	D₄
LCS	G ▷ C₂×C₉×C₁₇×C₃₇ ▷ C₉×C₁₇×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 22644

A minimal faithful permutation representation — 2 generators on 67 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,3,17,37,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^16,f^a=f^36,g^a=g^2

Conjugacy classes & character table

11325 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, 3, 17, 37, 3]);;
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.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D22644 #72277408