D₃₂₇₀₀ #72304209

order	65400
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₀₀

Splittings

Fitting	C₃×C₄×C₂₅×C₁₀₉ ⋊ C₂
Hall	C₃×C₂₅×C₁₀₉ ⋊ D₄
Hall	C₂₅×C₁₀₉ ⋊ D₁₂
Hall	C₃×C₁₀₉ ⋊ D₁₀₀
Sylow	C₁₀₉ ⋊ D₃₀₀
Hall	C₃×C₂₅ ⋊ D₄₃₆
Sylow	C₂₅ ⋊ D₁₃₀₈
Sylow	C₃ ⋊ D₁₀₉₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₈₁₇₅
commutator, cosocle	C₂×C₃×C₂₅×C₁₀₉	C₂²
Frattini	C₂×C₅	C₂×D₁₆₃₅
Fitting	C₃×C₄×C₂₅×C₁₀₉	C₂
socle	C₂×C₃×C₅×C₁₀₉	C₂×D₅
hypercenter	C₄	D₈₁₇₅
hypocenter	C₃×C₂₅×C₁₀₉	D₄
LCS	G ▷ C₂×C₃×C₂₅×C₁₀₉ ▷ C₃×C₂₅×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 32700

A minimal faithful permutation representation — 2 generators on 141 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);(2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2^3,3,5,109,5|b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^108,g^a=g^4

Conjugacy classes & character table

16353 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, 3, 5, 109, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^108);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32700 #72304209