C₇⋊C₁₆ #1128

order	112
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₄×AGL(1,𝔽₇) (order 336) · Inn(G) = D₇ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

C₇⋊C₁₆

C₇ ⋊ C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₇
commutator, hypocenter	C₇	C₁₆
Fitting, cosocle	C₇×C₈	C₂
socle, prime radical	C₂×C₇	C₈
LCS	G ▷ C₇
UCS	C₁ ◁ C₈

10 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:

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

Polycyclic presentation:

2^4,7|a^2=b,b^2=c,c^2=d,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 1);