D₂₅₄₀₀ #72288568

order	50800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₄₀₀

Splittings

Fitting	C₈×C₂₅×C₁₂₇ ⋊ C₂
Hall	C₂₅×C₁₂₇ ⋊ D₈
Sylow	C₁₂₇ ⋊ D₂₀₀
Sylow	C₂₅ ⋊ D₁₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₂₇₀₀
commutator, cosocle	C₄×C₂₅×C₁₂₇	C₂²
Frattini	C₄×C₅	C₂×D₆₃₅
Fitting	C₈×C₂₅×C₁₂₇	C₂
socle	C₂×C₅×C₁₂₇	D₂₀
hypercenter	C₈	D₃₁₇₅
hypocenter	C₂₅×C₁₂₇	D₈
LCS	G ▷ C₄×C₂₅×C₁₂₇ ▷ C₂×C₂₅×C₁₂₇ ▷ C₂₅×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 25400

A minimal faithful permutation representation — 2 generators on 160 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160);(2,8)(3,7)(4,6)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^4,5,127,5|b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^126,g^a=g^4

Conjugacy classes & character table

12703 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, 2, 5, 127, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25400 #72288568