D₂₂₆₂₄ #72277352

order	45248
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₆₂₄

Splittings

Fitting	C₇×C₃₂×C₁₀₁ ⋊ C₂
Hall	C₇×C₁₀₁ ⋊ D₃₂
Sylow	C₁₀₁ ⋊ D₂₂₄
Sylow	C₇ ⋊ D₃₂₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₁₃₁₂
commutator, cosocle	C₇×C₁₆×C₁₀₁	C₂²
Frattini	C₁₆	C₂×D₇₀₇
Fitting	C₇×C₃₂×C₁₀₁	C₂
socle	C₂×C₇×C₁₀₁	D₁₆
hypercenter	C₃₂	D₇₀₇
hypocenter	C₇×C₁₀₁	D₃₂
LCS	G ▷ C₇×C₁₆×C₁₀₁ ▷ C₇×C₈×C₁₀₁ ▷ C₄×C₇×C₁₀₁ ▷ C₂×C₇×C₁₀₁ ▷ C₇×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 140
transitive degree	μ_t(G) = 22624

A minimal faithful permutation representation — 2 generators on 140 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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,39)(35,38)(36,37)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)

Polycyclic presentation:

2^6,7,101|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6,h^a=h^100

Conjugacy classes & character table

11315 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 7, 101]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D22624 #72277352