Dic₃₂₁₉ #27166140

order	12876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₂₁₉
unambiguous semidirect product	C₃⋊Dic₁₀₇₃

Splittings

derived, Hall	C₃×C₂₉×C₃₇ ⋊ C₄
Hall	C₂₉×C₃₇ ⋊ Dic₃
Hall	C₃×C₃₇ ⋊ Dic₂₉
Hall	C₃×C₂₉ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₈₇
Sylow	C₂₉ ⋊ Dic₁₁₁
Sylow	C₃ ⋊ Dic₁₀₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₂₁₉
commutator, hypocenter	C₃×C₂₉×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₂₉×C₃₇	C₂
LCS	G ▷ C₃×C₂₉×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 12876

A minimal faithful permutation representation — 2 generators on 73 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);(1,3,2,4)(6,7)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,3,29,37,2|a^2=e,b^a=b^2,c^a=c^28,d^a=d^36

Conjugacy classes & character table

3222 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₆₅₇, C₃⋊(Dic₁₀₇₃⋊C₃), C₃⋊Dic₃₂₁₉, …
maximal normal subgroup of	C₃⋊(Dic₁₀₇₃⋊C₃), C₃×Dic₃₂₁₉, Dic₆₄₃₈
maximal quotient of	Dic₉₆₅₇, C₃⋊Dic₃₂₁₉, C₃×Dic₃₂₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 9);

Dic3219 #27166140