Dic₄₄₀₂ #72215859

order	17608
automorphism	Aut(G) order 36976800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₄₀₂

Splittings

Hall	C₃₁×C₇₁ ⋊ Q₈
Sylow	C₇₁ ⋊ Dic₆₂
Sylow	C₃₁ ⋊ Dic₁₄₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₂₀₁
commutator, socle, prime radical, cosocle	C₂×C₃₁×C₇₁	C₂²
Fitting	C₄×C₃₁×C₇₁	C₂
hypercenter	C₄	D₂₂₀₁
hypocenter	C₃₁×C₇₁	Q₈
LCS	G ▷ C₂×C₃₁×C₇₁ ▷ C₃₁×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 17608

A minimal faithful permutation representation — 2 generators on 110 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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110);(1,5,3,7)(2,6,4,8)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^3,31,71|a^2=c,b^2=c,b^a=bc,d^a=d^30,e^a=e^70

Conjugacy classes & character table

4405 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₀₄, Dic₁₃₂₀₆
maximal normal subgroup of	Dic₈₈₀₄
maximal quotient of	Dic₁₃₂₀₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 31, 71]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4402 #72215859