Dic₄₈₉₉ #34086677

order	19596
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₈₉₉
unambiguous semidirect product	C₃⋊Dic₁₆₃₃
unambiguous semidirect product	C₂₃⋊Dic₂₁₃
unambiguous semidirect product	C₇₁⋊Dic₆₉

Splittings

derived, Hall	C₃×C₂₃×C₇₁ ⋊ C₄
Hall	C₂₃×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₂₃
Sylow	C₇₁ ⋊ Dic₆₉
Hall	C₃×C₂₃ ⋊ Dic₇₁
Sylow	C₂₃ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₁₆₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₈₉₉
commutator, hypocenter	C₃×C₂₃×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₇₁	C₂
LCS	G ▷ C₃×C₂₃×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 19596

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

Polycyclic presentation:

2,3,23,71,2|a^2=e,b^a=b^2,c^a=c^22,d^a=d^70

Conjugacy classes & character table

4902 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₇₉₈, Dic₁₄₆₉₇
maximal normal subgroup of	Dic₉₇₉₈
maximal quotient of	Dic₁₄₆₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19596, 8);

Dic4899 #34086677