Dic₁₄₉₈ #72211981

order	5992
automorphism	Aut(G) order 3810912
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₈

Splittings

Hall	C₇×C₁₀₇ ⋊ Q₈
Sylow	C₁₀₇ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₂₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₉
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₀₇	C₂²
Fitting	C₄×C₇×C₁₀₇	C₂
hypercenter	C₄	D₇₄₉
hypocenter	C₇×C₁₀₇	Q₈
LCS	G ▷ C₂×C₇×C₁₀₇ ▷ C₇×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 5992

A minimal faithful permutation representation — 2 generators on 122 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^3,7,107|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^106

Conjugacy classes & character table

1501 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₉₉₆, Dic₄₄₉₄, Dic₇₄₉₀, …
maximal normal subgroup of	Dic₂₉₉₆
maximal quotient of	Dic₄₄₉₄, Dic₇₄₉₀, Dic₁₀₄₈₆

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, 7, 107]);;
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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1498 #72211981