C₇×#7490 #13184706

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#7490

Subgroups

	subgroup	G/H
center	C₄×C₇	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₇
Frattini, cosocle	#532	C₂²×C₇
socle	C₂×C₇	#1390
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

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

normal subgroups only

This group has 274 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)(8,9,11,16)(10,20,22,14)(12,17,27,34)(13,19,29,36)(15,30,18,28)(21,37,24,35)(23,38,26,31)(25,33,39,32);(8,10,12,23,13,25,30,24)(9,14,17,31,19,32,18,37)(11,26,27,39,29,21,28,22)(15,33,16,35,34,20,36,38)

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, 478);