Dic₄₄₃₃ #31801113

order	17732
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₄₃₃
unambiguous semidirect product	C₁₁⋊Dic₄₀₃
unambiguous semidirect product	C₃₁⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₃₁ ⋊ C₄
Hall	C₁₃×C₃₁ ⋊ Dic₁₁
Hall	C₁₁×C₃₁ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₃₄₁
Sylow	C₁₁ ⋊ Dic₄₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₄₃₃
commutator, hypocenter	C₁₁×C₁₃×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₃₁	C₂
LCS	G ▷ C₁₁×C₁₃×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 17732

A minimal faithful permutation representation — 2 generators on 59 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2,11,13,31,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^30

Conjugacy classes & character table

4436 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₆₆, Dic₁₃₂₉₉
maximal normal subgroup of	Dic₈₈₆₆
maximal quotient of	Dic₁₃₂₉₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17732, 8);

Dic4433 #31801113