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C₄×C₇ #85

order	28
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃ (order 12) · Inn(G) = C₁ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₈
abelian	C₄×C₇
multiplicative	(ℤ/29)ˣ, (ℤ/58)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₇
socle, prime radical	C₂×C₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 1 generator on 11 points:

(1,2,3,4)(5,6,7,8,9,10,11)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

center of	C₇×M₁₆, C₇×#41, C₇×C₃⋊C₈, …
derived subgroup of	#1132, D₅₆, Dic₂₈, …
Frattini subgroup of	C₄₉⋊C₈, C₈×C₄₉, Dic₁₉₆, …
Fitting subgroup of	C₇⋊C₈, Dic₁₄, C₄×D₇, …
maximal subgroup of	C₇⋊C₈, C₇×C₈, Dic₁₄, …
maximal normal subgroup of	C₇⋊C₈, C₇×C₈, Dic₁₄, …
maximal quotient of	C₇×C₈, C₂×C₄×C₇, C₇×Dic₃, …
automorphism group of	C₂₉, C₂×C₂₉
outer automorphism group of	C₂₉, C₂×C₂₉, D₃×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28, 2);