C₇×#7477 #13184693

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

Aliases

direct product

C₇×#7477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#357
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
prime radical	C₂³×C₇	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

178 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 43 points:

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

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, 465);