C₇×#7306 #13184522

order	1792
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7306

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#342
commutator	C₂⁴	C₄²×C₇
Frattini, cosocle	C₂²×#41	C₂²×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×#41 ◁ G

456 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,13)(12,15)(16,17,20,25,22,27,21,26)(18,28,30,43,32,45,31,44)(19,33,34,42,35,24,36,40)(23,37,38,29,39,46,41,47);(8,10)(9,12,13,14)(11,15)(16,18,21,31,22,32,20,30)(17,23,26,38,27,39,25,41)(19,29,34,46,35,47,36,37)(24,28,40,44,33,45,42,43)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 294);