Dic₁₄₀₄₂ #72290451

order	56168
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₀₄₂

Splittings

Hall	C₇×C₁₇×C₅₉ ⋊ Q₈
Hall	C₁₇×C₅₉ ⋊ Dic₁₄
Hall	C₇×C₅₉ ⋊ Dic₃₄
Hall	C₇×C₁₇ ⋊ Dic₁₁₈
Sylow	C₅₉ ⋊ Dic₂₃₈
Sylow	C₁₇ ⋊ Dic₈₂₆
Sylow	C₇ ⋊ Dic₂₀₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₀₂₁
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₅₉	C₂²
Fitting	C₄×C₇×C₁₇×C₅₉	C₂
hypercenter	C₄	D₇₀₂₁
hypocenter	C₇×C₁₇×C₅₉	Q₈
LCS	G ▷ C₂×C₇×C₁₇×C₅₉ ▷ C₇×C₁₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 56168

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

Polycyclic presentation:

2^3,7,17,59|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^16,f^a=f^58

Conjugacy classes & character table

14045 conjugacy classes. The character table is too large to display.

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, 7, 17, 59]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14042 #72290451