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C₅×C₇ #148

order	35
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₄ (order 24) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₅
abelian	C₅×C₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 35

A minimal faithful permutation representation — 1 generator on 12 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

center of	C₅×D₃×C₇, C₅×C₇×D₅, C₅×C₇×A₄, …
derived subgroup of	D₃₅, Dic₃₅, C₇⋊AGL(1,𝔽₅), …
Frattini subgroup of	C₂₅×C₄₉, C₂×C₂₅×C₄₉, C₄₉×D₂₅, …
Fitting subgroup of	C₇×D₅, C₅×D₇, D₃₅, …
socle of	C₇×D₅, C₅×D₇, D₃₅, …
maximal subgroup of	C₇×D₅, C₅×D₇, D₃₅, …
maximal normal subgroup of	C₇×D₅, C₅×D₇, D₃₅, …
maximal quotient of	C₂×C₅×C₇, C₃×C₅×C₇, C₇×C₂₅, …
outer automorphism group of	D₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35, 1);