C₇×#7479 #13184695

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7479

Subgroups

	subgroup	G/H
center	C₄×C₇	#357
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂×C₇	#1380
prime radical	C₂³×C₇	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

162 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,18,15,22,35,21)(10,23,25,55,28,46,60,56)(11,29,32,52,33,19,36,50)(13,17,34,47,37,49,68,48)(14,38,31,20,39,51,67,53)(16,40,41,30,45,66,44,24)(26,61,59,71,58,54,69,43)(27,62,57,42,63,70,65,64);(8,10,13,26,14,27,36,40,35,60,68,69,67,65,33,24,15,28,37,58,39,63,32,66,12,25,34,59,31,57,11,30)(9,16,19,42,20,43,48,55,18,41,50,70,51,61,17,46,22,45,29,64,53,71,47,56,21,44,52,62,38,54,49,23)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 467);