Dic₄₇₃₁ #34038726

order	18924
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₇₃₁
unambiguous semidirect product	C₃⋊Dic₁₅₇₇
unambiguous semidirect product	C₁₉⋊Dic₂₄₉
unambiguous semidirect product	C₈₃⋊Dic₅₇

Splittings

derived, Hall	C₃×C₁₉×C₈₃ ⋊ C₄
Hall	C₁₉×C₈₃ ⋊ Dic₃
Hall	C₃×C₈₃ ⋊ Dic₁₉
Sylow	C₈₃ ⋊ Dic₅₇
Hall	C₃×C₁₉ ⋊ Dic₈₃
Sylow	C₁₉ ⋊ Dic₂₄₉
Sylow	C₃ ⋊ Dic₁₅₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₇₃₁
commutator, hypocenter	C₃×C₁₉×C₈₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₈₃	C₂
LCS	G ▷ C₃×C₁₉×C₈₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 18924

A minimal faithful permutation representation — 2 generators on 109 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3,2,4)(6,7)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2,3,19,83,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^82

Conjugacy classes & character table

4734 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₄₆₂, Dic₁₄₁₉₃
maximal normal subgroup of	Dic₉₄₆₂
maximal quotient of	Dic₁₄₁₉₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18924, 9);

Dic4731 #34038726