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D₇×#106 #89631

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₇×#106 ⋊ C₂
Sylow	C₇ ⋊ C₂×#106

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#106	D₇
hypocenter	C₇	C₂×#106
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

(5,6,7,9,8,10,11,12)(13,14,16,18,19,17,15);(1,2,3,4)(6,9)(7,11)(10,12);(14,15)(16,17)(18,19)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#360, D₇×#365, C₂×D₇×#106, …
maximal normal subgroup of	D₇×#360, D₇×#365, C₂×D₇×#106, …
maximal quotient of	D₇×#334, #11726283, #11726286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 393);