C₇×#13306 #13190522

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13306

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

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

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, 6294);