Dic₄₀₄₇ #30633682

order	16188
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₄₇
unambiguous semidirect product	C₃⋊Dic₁₃₄₉
unambiguous semidirect product	C₁₉⋊Dic₂₁₃
unambiguous semidirect product	C₇₁⋊Dic₅₇

Splittings

derived, Hall	C₃×C₁₉×C₇₁ ⋊ C₄
Hall	C₁₉×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₁₉
Sylow	C₇₁ ⋊ Dic₅₇
Hall	C₃×C₁₉ ⋊ Dic₇₁
Sylow	C₁₉ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₁₃₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₄₇
commutator, hypocenter	C₃×C₁₉×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₇₁	C₂
LCS	G ▷ C₃×C₁₉×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 16188

A minimal faithful permutation representation — 2 generators on 97 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,3,2,4)(6,7)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(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:

2,3,19,71,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^70

Conjugacy classes & character table

4050 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₉₄
maximal normal subgroup of	Dic₈₀₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16188, 9);

Dic4047 #30633682