C₇×D₈ #1151

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) = C₂

Aliases

C₇×D₈

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

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 24);