C₇×#13574 #13190790

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13574

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

486 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,12,17,14,10,19,21)(11,15,22,20,13,18,23,16);(8,10)(9,14)(11,15)(12,16)(13,18)(17,23)(19,20)(21,22);(8,11,14,13)(9,15,10,18)(12,22,19,23)(16,17,20,21)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6562);