D₂₇₂₂₄ #72290547

order	54448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₂₂₄

Splittings

Fitting	C₈×C₄₁×C₈₃ ⋊ C₂
Hall	C₄₁×C₈₃ ⋊ D₈
Sylow	C₈₃ ⋊ D₃₂₈
Sylow	C₄₁ ⋊ D₆₆₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₁₂
commutator, cosocle	C₄×C₄₁×C₈₃	C₂²
Frattini	C₄	C₂×D₃₄₀₃
Fitting	C₈×C₄₁×C₈₃	C₂
socle	C₂×C₄₁×C₈₃	D₄
hypercenter	C₈	D₃₄₀₃
hypocenter	C₄₁×C₈₃	D₈
LCS	G ▷ C₄×C₄₁×C₈₃ ▷ C₂×C₄₁×C₈₃ ▷ C₄₁×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 27224

A minimal faithful permutation representation — 2 generators on 132 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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132);(2,8)(3,7)(4,6)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2^4,41,83|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^40,f^a=f^82

Conjugacy classes & character table

13615 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, 2, 41, 83]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^40);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27224 #72290547