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D₄×#261 #88239

order	416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	D₄×C₁₃ ⋊ C₄
Sylow	C₁₃ ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#261
commutator, socle	C₂×C₁₃	C₂²×C₄
Fitting	D₄×C₁₃	C₄
prime radical	D₄×D₁₃	C₂
cosocle	C₂×D₁₃	C₂³
hypercenter	D₄	#261
hypocenter	C₁₃	C₄×D₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ D₄

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 3 generators on 17 points:

(2,3,4,6)(5,8,9,13)(7,11,12,10);(1,2,5,8,12,6,10,11,3,7,13,9,4)(15,17);(14,15)(16,17)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₈×#261, #11719102, SD₁₆×#261, …
maximal normal subgroup of	D₈×#261, #11719102, SD₁₆×#261, …
maximal quotient of	#11719063, #11719064, #11719065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 206);