Dic₂₆₅₀ #72212866

order	10600
automorphism	Aut(G) order 11024000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₆₅₀

Splittings

Hall	C₂₅×C₅₃ ⋊ Q₈
Sylow	C₅₃ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₁₀₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₃₂₅
commutator, cosocle	C₂×C₂₅×C₅₃	C₂²
Frattini	C₂×C₅	C₂×D₂₆₅
Fitting	C₄×C₂₅×C₅₃	C₂
socle, prime radical	C₂×C₅×C₅₃	C₂×D₅
hypercenter	C₄	D₁₃₂₅
hypocenter	C₂₅×C₅₃	Q₈
LCS	G ▷ C₂×C₂₅×C₅₃ ▷ C₂₅×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 10600

A minimal faithful permutation representation — 2 generators on 86 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86);(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)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^3,5,53,5|a^2=c,b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^52,f^a=f^4

Conjugacy classes & character table

2653 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₃₀₀, Dic₇₉₅₀, Dic₁₃₂₅₀
maximal normal subgroup of	Dic₅₃₀₀
maximal quotient of	Dic₇₉₅₀, Dic₁₃₂₅₀

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, 53, 5]);;
gap> SetPower(coll, 1, F.3);;
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.5^52);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2650 #72212866