C₇×#7419 #13184635

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7419

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#357
commutator	C₂×C₁₆	C₂×C₄×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
prime radical	C₂³×C₇	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,14,15,12)(11,13)(17,23,24,39)(18,28)(19,34)(20,30)(21,36)(25,42,26,43)(27,44,38,41)(29,35,40,37)(31,33)(32,47)(45,46);(8,10)(9,12)(11,15)(13,14)(16,17,22,24)(18,29,33,40)(19,35,20,37)(21,38,36,27)(23,28,39,31)(25,30,26,34)(32,42,45,43)(41,47,44,46)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 407);