Dic₁₄₃₅₀ #72290977

order	57400
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₅₀

Splittings

Hall	C₇×C₂₅×C₄₁ ⋊ Q₈
Hall	C₂₅×C₄₁ ⋊ Dic₁₄
Hall	C₇×C₄₁ ⋊ Dic₅₀
Hall	C₇×C₂₅ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₃₅₀
Sylow	C₂₅ ⋊ Dic₅₇₄
Sylow	C₇ ⋊ Dic₂₀₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₇₅
commutator, cosocle	C₂×C₇×C₂₅×C₄₁	C₂²
Frattini	C₂×C₅	C₂×D₁₄₃₅
Fitting	C₄×C₇×C₂₅×C₄₁	C₂
socle, prime radical	C₂×C₅×C₇×C₄₁	C₂×D₅
hypercenter	C₄	D₇₁₇₅
hypocenter	C₇×C₂₅×C₄₁	Q₈
LCS	G ▷ C₂×C₇×C₂₅×C₄₁ ▷ C₇×C₂₅×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 57400

A minimal faithful permutation representation — 2 generators on 81 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(35,40)(36,39)(37,38)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2^3,5,7,41,5|a^2=c,b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^6,f^a=f^40,g^a=g^4

Conjugacy classes & character table

14353 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, 41, 5]);;
gap> SetPower(coll, 1, F.3);;
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.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14350 #72290977