D₂₁₇₀₀ #72275223

order	43400
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₇₀₀

Splittings

Fitting	C₄×C₇×C₂₅×C₃₁ ⋊ C₂
Hall	C₇×C₂₅×C₃₁ ⋊ D₄
Hall	C₂₅×C₃₁ ⋊ D₂₈
Hall	C₇×C₃₁ ⋊ D₁₀₀
Hall	C₇×C₂₅ ⋊ D₁₂₄
Sylow	C₃₁ ⋊ D₇₀₀
Sylow	C₂₅ ⋊ D₈₆₈
Sylow	C₇ ⋊ D₃₁₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₄₂₅
commutator, cosocle	C₂×C₇×C₂₅×C₃₁	C₂²
Frattini	C₂×C₅	C₂×D₁₀₈₅
Fitting	C₄×C₇×C₂₅×C₃₁	C₂
socle	C₂×C₅×C₇×C₃₁	C₂×D₅
hypercenter	C₄	D₅₄₂₅
hypocenter	C₇×C₂₅×C₃₁	D₄
LCS	G ▷ C₂×C₇×C₂₅×C₃₁ ▷ C₇×C₂₅×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 21700

A minimal faithful permutation representation — 2 generators on 67 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);(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)(31,36)(32,35)(33,34)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^3,5,7,31,5|b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^6,f^a=f^30,g^a=g^4

Conjugacy classes & character table

10853 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, 5, 7, 31, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D21700 #72275223