D₃₀₆₃₆ #72295955

order	61272
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₆₃₆

Splittings

Fitting	C₄×C₉×C₂₃×C₃₇ ⋊ C₂
Hall	C₉×C₂₃×C₃₇ ⋊ D₄
Hall	C₂₃×C₃₇ ⋊ D₃₆
Hall	C₉×C₃₇ ⋊ D₉₂
Hall	C₉×C₂₃ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₈₂₈
Sylow	C₂₃ ⋊ D₁₃₃₂
Sylow	C₉ ⋊ D₃₄₀₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₆₅₉
commutator, cosocle	C₂×C₉×C₂₃×C₃₇	C₂²
Frattini	C₂×C₃	C₂×D₂₅₅₃
Fitting	C₄×C₉×C₂₃×C₃₇	C₂
socle	C₂×C₃×C₂₃×C₃₇	C₂×D₃
hypercenter	C₄	D₇₆₅₉
hypocenter	C₉×C₂₃×C₃₇	D₄
LCS	G ▷ C₂×C₉×C₂₃×C₃₇ ▷ C₉×C₂₃×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 30636

A minimal faithful permutation representation — 2 generators on 73 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^3,3,23,37,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^22,f^a=f^36,g^a=g^2

Conjugacy classes & character table

15321 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, 23, 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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30636 #72295955