C₂×Dic₅₇ #90717

order	456
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×Dic₅₇
unambiguous semidirect product	(C₂×C₃)⋊Dic₁₉
unambiguous semidirect product	(C₂×C₁₉)⋊Dic₃

Splittings

derived, Hall	C₃×C₁₉ ⋊ C₂×C₄
Sylow	C₁₉ ⋊ C₂×Dic₃
Sylow	C₃ ⋊ C₂×Dic₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₅₇
commutator, hypocenter	C₃×C₁₉	C₂×C₄
Frattini	C₂	C₂×D₅₇
Fitting, socle, prime radical	C₂²×C₃×C₁₉	C₂
cosocle	C₂×C₃×C₁₉	C₂²
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₂²

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 456

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2)(4,5)(6,7,8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4,5)(10,11,13,15,17,19,21,23,25,27,28,26,24,22,20,18,16,14,12)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×Dic₁₉, (C₂×D₁₉)⋊Dic₃, (C₂×D₃)⋊Dic₁₉, …
maximal normal subgroup of	Dic₃×Dic₁₉, (C₂×D₁₉)⋊Dic₃, (C₂×D₃)⋊Dic₁₉, …
maximal quotient of	C₂×C₃⋊(C₁₉⋊C₈), #11744885, C₄×Dic₅₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 37);

C2×Dic57 #90717

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×Dic₅₇ #90717