Dic₃₅₀₂ #72214280

order	14008
automorphism	Aut(G) order 22861056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₅₀₂

Splittings

Hall	C₁₇×C₁₀₃ ⋊ Q₈
Sylow	C₁₀₃ ⋊ Dic₃₄
Sylow	C₁₇ ⋊ Dic₂₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₇₅₁
commutator, socle, prime radical, cosocle	C₂×C₁₇×C₁₀₃	C₂²
Fitting	C₄×C₁₇×C₁₀₃	C₂
hypercenter	C₄	D₁₇₅₁
hypocenter	C₁₇×C₁₀₃	Q₈
LCS	G ▷ C₂×C₁₇×C₁₀₃ ▷ C₁₇×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 14008

A minimal faithful permutation representation — 2 generators on 128 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(1,5,3,7)(2,6,4,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^3,17,103|a^2=c,b^2=c,b^a=bc,d^a=d^16,e^a=e^102

Conjugacy classes & character table

3505 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₀₀₄, Dic₁₀₅₀₆
maximal normal subgroup of	Dic₇₀₀₄
maximal quotient of	Dic₁₀₅₀₆

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, 17, 103]);;
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.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3502 #72214280