D₁₃₁₀₀ #72235355

order	26200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₁₀₀

Splittings

Fitting	C₄×C₂₅×C₁₃₁ ⋊ C₂
Hall	C₂₅×C₁₃₁ ⋊ D₄
Sylow	C₁₃₁ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₅₂₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₂₇₅
commutator, cosocle	C₂×C₂₅×C₁₃₁	C₂²
Frattini	C₂×C₅	C₂×D₆₅₅
Fitting	C₄×C₂₅×C₁₃₁	C₂
socle	C₂×C₅×C₁₃₁	C₂×D₅
hypercenter	C₄	D₃₂₇₅
hypocenter	C₂₅×C₁₃₁	D₄
LCS	G ▷ C₂×C₂₅×C₁₃₁ ▷ C₂₅×C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 13100

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);(2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)

Polycyclic presentation:

2^3,5,131,5|b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^130,f^a=f^4

Conjugacy classes & character table

6553 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₆₂₀₀
maximal normal subgroup of	D₂₆₂₀₀
maximal quotient of	Dic₁₃₁₀₀, D₂₆₂₀₀

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, 5, 131, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^130);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13100 #72235355