Dic₃₀₂₁ #27130577

order	12084
automorphism	Aut(G) order 11310624
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₀₂₁
unambiguous semidirect product	C₃⋊Dic₁₀₀₇
unambiguous semidirect product	C₁₉⋊Dic₁₅₉

Splittings

derived, Hall	C₃×C₁₉×C₅₃ ⋊ C₄
Hall	C₁₉×C₅₃ ⋊ Dic₃
Hall	C₃×C₅₃ ⋊ Dic₁₉
Hall	C₃×C₁₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₅₇
Sylow	C₁₉ ⋊ Dic₁₅₉
Sylow	C₃ ⋊ Dic₁₀₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₀₂₁
commutator, hypocenter	C₃×C₁₉×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₅₃	C₂
LCS	G ▷ C₃×C₁₉×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 12084

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

Polycyclic presentation:

2,3,19,53,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^52

Conjugacy classes & character table

3024 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₆₃, C₃⋊((C₁₉⋊C₃)⋊Dic₅₃), C₃⋊Dic₃₀₂₁, …
maximal normal subgroup of	C₃⋊((C₁₉⋊C₃)⋊Dic₅₃), C₃×Dic₃₀₂₁, Dic₆₀₄₂
maximal quotient of	Dic₉₀₆₃, C₃⋊Dic₃₀₂₁, C₃×Dic₃₀₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12084, 9);

Dic3021 #27130577