C₂×C₇×D₈ #6535

order	224
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×C₃×#23882 (order 1536) · Inn(G) = D₄ · Out(G) = C₂×C₃×#120
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₇×D₈

	subgroup	G/H
center, socle	C₂²×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ G

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2)(11,12,13,14,15,16,17);(3,4,5,6,7,8,9,10);(4,10)(5,9)(6,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 167);