C₇×#7488 #13184704

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7488

Subgroups

	subgroup	G/H
center	C₄×C₇	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄²×C₇
Frattini, cosocle	#532	C₂²×C₇
socle	C₂×C₇	#1390
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 224

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

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 476);