C₇×#12133 #13189349

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12090
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₇×#1595	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₂²⋊C₄ ◁ G

554 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5121);