order	224
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = C₂×C₃×#442 (order 384) · Inn(G) = D₈ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

C₇×SD₃₂

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	D₈
commutator, Frattini, cosocle	C₈	C₂²×C₇
prime radical	C₇×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ G

26 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)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(9,15)(10,22)(11,13)(12,20)(14,18)(17,23)(19,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 61);