D₂₇₄₀₀ #72290708

order	54800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₄₀₀

Splittings

Fitting	C₈×C₂₅×C₁₃₇ ⋊ C₂
Hall	C₂₅×C₁₃₇ ⋊ D₈
Sylow	C₁₃₇ ⋊ D₂₀₀
Sylow	C₂₅ ⋊ D₁₀₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₃₇₀₀
commutator, cosocle	C₄×C₂₅×C₁₃₇	C₂²
Frattini	C₄×C₅	C₂×D₆₈₅
Fitting	C₈×C₂₅×C₁₃₇	C₂
socle	C₂×C₅×C₁₃₇	D₂₀
hypercenter	C₈	D₃₄₂₅
hypocenter	C₂₅×C₁₃₇	D₈
LCS	G ▷ C₄×C₂₅×C₁₃₇ ▷ C₂×C₂₅×C₁₃₇ ▷ C₂₅×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 170
transitive degree	μ_t(G) = 27400

A minimal faithful permutation representation — 2 generators on 170 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170);(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)(35,170)(36,169)(37,168)(38,167)(39,166)(40,165)(41,164)(42,163)(43,162)(44,161)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2^4,5,137,5|b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^136,g^a=g^4

Conjugacy classes & character table

13703 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, 137, 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.6^136);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27400 #72290708