C₇×#7430 #13184646

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7430

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₇
Frattini, cosocle	#390	C₂²×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#390 ◁ G

236 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 418);