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C₂×C₇² #1030

order	98
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

C₂×C₇²

Subgroups

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

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 98

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

center of	C₇²×D₄, C₇²×Q₈, C₇²×Dic₃, …
derived subgroup of	#87640, #87641, #87642, …
Fitting subgroup of	C₇×Dic₇, C₇⋊Dic₇, C₂×C₇×D₇, …
socle of	C₇×Dic₇, C₇⋊Dic₇, C₄×C₇², …
maximal subgroup of	C₇×Dic₇, C₇⋊Dic₇, C₄×C₇², …
maximal normal subgroup of	C₇×Dic₇, C₇⋊Dic₇, C₄×C₇², …
maximal quotient of	C₄×C₇², C₂²×C₇², D₃×C₇², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(98, 5);