C₇⋊C₃₂ #6369

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₈×AGL(1,𝔽₇) (order 672) · Inn(G) = D₇ · Out(G) = C₂×C₃×C₈
homology	Abelianization (H₁) = C₃₂ · Schur multiplier (H₂) = C₁

Aliases

C₇⋊C₃₂

C₇ ⋊ C₃₂

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	D₇
commutator, hypocenter	C₇	C₃₂
Fitting, cosocle	C₇×C₁₆	C₂
socle, prime radical	C₂×C₇	C₁₆
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₆

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(2,3)(4,5)(6,7)(8,9,10,14,11,15,18,24,12,16,19,25,21,27,30,34,13,17,20,26,22,28,31,35,23,29,32,36,33,37,38,39);(1,2,4,6,7,5,3)

Polycyclic presentation:

2^5,7|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 1);