Dic₄₁₉₉ #30680990

order	16796
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₁₉₉
unambiguous semidirect product	C₁₉⋊Dic₂₂₁

Splittings

derived, Hall	C₁₃×C₁₇×C₁₉ ⋊ C₄
Hall	C₁₇×C₁₉ ⋊ Dic₁₃
Hall	C₁₃×C₁₉ ⋊ Dic₁₇
Hall	C₁₃×C₁₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₂₂₁
Sylow	C₁₇ ⋊ Dic₂₄₇
Sylow	C₁₃ ⋊ Dic₃₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₁₉₉
commutator, hypocenter	C₁₃×C₁₇×C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₁₇×C₁₉	C₂
LCS	G ▷ C₁₃×C₁₇×C₁₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 16796

A minimal faithful permutation representation — 2 generators on 53 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2,13,17,19,2|a^2=e,b^a=b^12,c^a=c^16,d^a=d^18

Conjugacy classes & character table

4202 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₃₉₈, Dic₁₂₅₉₇
maximal normal subgroup of	Dic₈₃₉₈
maximal quotient of	Dic₁₂₅₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16796, 8);

Dic4199 #30680990