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C₂×Dic₅₁ #87991

order	408
isoclinism	isoclinic class of D₅₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×Dic₅₁
unambiguous semidirect product	(C₂×C₃)⋊Dic₁₇

Splittings

derived, Hall	C₃×C₁₇ ⋊ C₂×C₄
Sylow	C₁₇ ⋊ C₂×Dic₃
Sylow	C₃ ⋊ C₂×Dic₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₅₁
commutator, hypocenter	C₃×C₁₇	C₂×C₄
Frattini	C₂	C₂×D₅₁
Fitting, socle, prime radical	C₂²×C₃×C₁₇	C₂
cosocle	C₂×C₃×C₁₇	C₂²
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₂²

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 408

A minimal faithful permutation representation — 2 generators on 26 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);(1,2)(3,4,5)(10,11,13,15,17,19,21,23,25,26,24,22,20,18,16,14,12)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×Dic₁₇, #11717756, (C₂×D₃)⋊Dic₁₇, …
maximal normal subgroup of	Dic₃×Dic₁₇, #11717756, (C₂×D₃)⋊Dic₁₇, …
maximal quotient of	C₂×C₃⋊#3622, #11717801, C₄×Dic₅₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 28);