C₇×#7047 #13184263

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7047

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#95
commutator	C₂²×C₄	C₄²×C₇
Frattini, cosocle	C₂⁴×C₄	C₂²×C₇
prime radical	C₂⁵×C₇	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

608 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 39
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 39 points:

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

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, 35);