C₇×#7055 #13184271

order	1792
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7055

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#99
commutator	C₂³	C₂×C₇×C₁₆
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₇
socle	C₂²×C₇	#323
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 448

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

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 43);