C₇×#7428 #13184644

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7428

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#354
commutator	C₂×C₄²	C₂×C₄×C₇
Frattini, cosocle	#390	C₂²×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#390 ◁ G

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

normal subgroups only

This group has 226 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) = 448

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 416);