Dic₅₇ #6579

order	228
isoclinism	isoclinic class of D₅₇
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₉) (order 4104) · Inn(G) = D₅₇ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇
unambiguous semidirect product	C₃⋊Dic₁₉
unambiguous semidirect product	C₁₉⋊Dic₃

Splittings

derived, Hall	C₃×C₁₉ ⋊ C₄
Sylow	C₁₉ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇
commutator, hypocenter	C₃×C₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 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);(1,3,2,4)(6,7)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×D₁₉, D₃×Dic₁₉, #90695, …
maximal normal subgroup of	Dic₃×D₁₉, D₃×Dic₁₉, #90695, …
maximal quotient of	C₃⋊(C₁₉⋊C₈), C₂×Dic₅₇, Dic₁₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(228, 5);

Dic57 #6579

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

Dic₅₇ #6579