Dic₂₆₇₈ #72212896

order	10712
automorphism	Aut(G) order 13111488
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₆₇₈

Splittings

Hall	C₁₃×C₁₀₃ ⋊ Q₈
Sylow	C₁₀₃ ⋊ Dic₂₆
Sylow	C₁₃ ⋊ Dic₂₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃₃₉
commutator, socle, prime radical, cosocle	C₂×C₁₃×C₁₀₃	C₂²
Fitting	C₄×C₁₃×C₁₀₃	C₂
hypercenter	C₄	D₁₃₃₉
hypocenter	C₁₃×C₁₀₃	Q₈
LCS	G ▷ C₂×C₁₃×C₁₀₃ ▷ C₁₃×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 10712

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

Polycyclic presentation:

2^3,13,103|a^2=c,b^2=c,b^a=bc,d^a=d^12,e^a=e^102

Conjugacy classes & character table

2681 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₃₅₆, Dic₈₀₃₄, Dic₁₃₃₉₀
maximal normal subgroup of	Dic₅₃₅₆
maximal quotient of	Dic₈₀₃₄, Dic₁₃₃₉₀

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, 13, 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.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2678 #72212896