C₂×C₇×Q₁₆ #6537

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) = C₂²

Aliases

C₂×C₇×Q₁₆

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

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2)(19,20,21,22,23,24,25);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 169);