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C₂×#6407 #89988

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

Aliases

Splittings

C₇ ⋊ C₂×#100

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂²×C₇×D₄	C₂
socle	C₂²×C₇	C₂²⋊C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×D₄	D₇
hypocenter	C₇	C₂×#100
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

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

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725443, #11725500, #11727396, …
maximal normal subgroup of	#11725443, #11725500, #11727396, …
maximal quotient of	C₂×#89320, #11729293, C₂×#89325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 750);