Dic₅₁₅₉ #35207492

order	20636
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₅₉
unambiguous semidirect product	C₇⋊Dic₇₃₇
unambiguous semidirect product	C₁₁⋊Dic₄₆₉
unambiguous semidirect product	C₆₇⋊Dic₇₇

Splittings

derived, Hall	C₇×C₁₁×C₆₇ ⋊ C₄
Hall	C₁₁×C₆₇ ⋊ Dic₇
Hall	C₇×C₆₇ ⋊ Dic₁₁
Hall	C₇×C₁₁ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₇₇
Sylow	C₁₁ ⋊ Dic₄₆₉
Sylow	C₇ ⋊ Dic₇₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₅₉
commutator, hypocenter	C₇×C₁₁×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₆₇	C₂
LCS	G ▷ C₇×C₁₁×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 20636

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(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);(1,3,2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,7,11,67,2|a^2=e,b^a=b^6,c^a=c^10,d^a=d^66

Conjugacy classes & character table

5162 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₁₈, Dic₁₅₄₇₇
maximal normal subgroup of	Dic₁₀₃₁₈
maximal quotient of	Dic₁₅₄₇₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20636, 9);

Dic5159 #35207492