D₂₇₁₅₆ #72290482

order	54312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₅₆

Splittings

Fitting	C₃×C₄×C₃₁×C₇₃ ⋊ C₂
Hall	C₃×C₃₁×C₇₃ ⋊ D₄
Hall	C₃₁×C₇₃ ⋊ D₁₂
Hall	C₃×C₇₃ ⋊ D₁₂₄
Hall	C₃×C₃₁ ⋊ D₂₉₂
Sylow	C₇₃ ⋊ D₃₇₂
Sylow	C₃₁ ⋊ D₈₇₆
Sylow	C₃ ⋊ D₉₀₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₇₈₉
commutator, socle, cosocle	C₂×C₃×C₃₁×C₇₃	C₂²
Fitting	C₃×C₄×C₃₁×C₇₃	C₂
hypercenter	C₄	D₆₇₈₉
hypocenter	C₃×C₃₁×C₇₃	D₄
LCS	G ▷ C₂×C₃×C₃₁×C₇₃ ▷ C₃×C₃₁×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 27156

A minimal faithful permutation representation — 2 generators on 111 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,4)(6,7)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^3,3,31,73|b^2=c,b^a=bc,d^a=d^2,e^a=e^30,f^a=f^72

Conjugacy classes & character table

13581 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 31, 73]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27156 #72290482