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D₇×#97 #89476

order	448
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₇×#97 ⋊ C₂
Sylow	C₇ ⋊ C₂×#97

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₇×#97	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	C₂×#97
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

146 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#329, #11726095, #11726099, …
maximal normal subgroup of	D₇×#329, #11726095, #11726099, …
maximal quotient of	D₇×#322, #11725823, #11725824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 238);