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C₈×Dic₇ #6387

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×#120×AGL(1,𝔽₇) (order 2688) · Inn(G) = D₇ · Out(G) = C₂×C₃×#120
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

C₈×Dic₇

Splittings

C₇ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	D₇
commutator, hypocenter	C₇	C₄×C₈
Frattini	C₂×C₄	C₂×D₇
Fitting	C₂×C₇×C₈	C₂
socle, prime radical	C₂²×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂²
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×Dic₇, Dic₇⋊C₁₆, #89297, …
maximal normal subgroup of	C₁₆×Dic₇, Dic₇⋊C₁₆, #89297, …
maximal quotient of	C₈×C₇⋊C₈, #89249, C₁₆×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 19);