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C₇² #252

order	49
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×#10621462 (order 2016) · Inn(G) = C₁ · Out(G) = C₃×#10621462
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₇

Aliases

C₇²

Subgroups

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

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 49

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₇², C₇²×D₅, C₇²×A₄, …
derived subgroup of	C₇⋊D₇, #3861, #3862, …
Frattini subgroup of	C₄₉², C₇²⋊C₄₉, C₄₉⋊C₄₉, …
Fitting subgroup of	C₇×D₇, C₇⋊D₇, C₇×ΓL(1,𝔽₈), …
socle of	C₇×D₇, C₇⋊D₇, C₇×ΓL(1,𝔽₈), …
maximal subgroup of	C₇×D₇, C₇⋊D₇, C₂×C₇², …
maximal normal subgroup of	C₇×D₇, C₇⋊D₇, C₂×C₇², …
maximal quotient of	C₂×C₇², C₃×C₇², C₅×C₇², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(49, 2);