H(7,𝔽₂)⋊C₇ #11744543

order	896
automorphism	?
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	#90632
commutator, Fitting, cosocle, hypocenter	H(7,𝔽₂)	C₇
LCS	G ▷ H(7,𝔽₂)
UCS	C₁ ◁ C₂

441 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

Polycyclic presentation:

7,2^7|b^2=h,c^2=h,d^2=h,b^a=bd,c^a=bceh,d^a=cdh,e^a=cegh,f^a=defg,g^a=def,c^b=ch,d^b=dh,e^b=eh,g^b=gh,d^c=dh,e^c=eh,f^c=fh,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 19309);