C₇×#7192 #13184408

order	1792
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7192

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#99
commutator	C₂³	C₄×C₇×C₈
Frattini, cosocle	C₂²×M₁₆	C₂²×C₇
socle	C₂²×C₇	#324
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×M₁₆ ◁ G

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3,4,5,6,7)(12,13,15,21,17,23,31,39,18,20,29,36,16,22,30,38)(14,24,25,37,27,34,41,42,28,19,32,33,26,35,40,43);(8,9,10,11)(12,14,16,26,18,28,17,27)(13,19,22,34,20,24,23,35)(15,25,30,40,29,32,31,41)(21,37,38,43,36,33,39,42)

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, 180);