order	112
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×C₃×AGL(1,ℤ/8) (order 192) · Inn(G) = D₄ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

C₇×Q₁₆

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ G

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15)(16,23,22,21,20,19,18,17);(8,16,12,20)(9,17,13,21)(10,18,14,22)(11,19,15,23)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 26);