Dic₄₃₇₁ #31775719

order	17484
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₃₇₁
unambiguous semidirect product	C₃⋊Dic₁₄₅₇
unambiguous semidirect product	C₃₁⋊Dic₁₄₁
unambiguous semidirect product	C₄₇⋊Dic₉₃

Splittings

derived, Hall	C₃×C₃₁×C₄₇ ⋊ C₄
Hall	C₃₁×C₄₇ ⋊ Dic₃
Hall	C₃×C₄₇ ⋊ Dic₃₁
Hall	C₃×C₃₁ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₉₃
Sylow	C₃₁ ⋊ Dic₁₄₁
Sylow	C₃ ⋊ Dic₁₄₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₃₇₁
commutator, hypocenter	C₃×C₃₁×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₁×C₄₇	C₂
LCS	G ▷ C₃×C₃₁×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 17484

A minimal faithful permutation representation — 2 generators on 85 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);(1,3,2,4)(6,7)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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:

2,3,31,47,2|a^2=e,b^a=b^2,c^a=c^30,d^a=d^46

Conjugacy classes & character table

4374 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₄₂, Dic₁₃₁₁₃
maximal normal subgroup of	Dic₈₇₄₂
maximal quotient of	Dic₁₃₁₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17484, 9);

Dic4371 #31775719