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D₇×SD₁₆ #6476

order	224
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₇) (order 1344) · Inn(G) = D₄×D₇ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₇×SD₁₆

Splittings

Fitting	C₇×SD₁₆ ⋊ C₂
Sylow	C₇ ⋊ C₂×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄×D₇
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₇×SD₁₆	C₂
socle	C₂×C₇	C₂×D₄
prime radical	D₄×D₇	C₂
hypercenter	SD₁₆	D₇
hypocenter	C₇	C₂×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₇×SD₁₆, #90453, D₇×#135, …
maximal normal subgroup of	C₂×D₇×SD₁₆, #90453, D₇×#135, …
maximal quotient of	#89530, #89532, #89533, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 108);