C₇×#13569 #13190785

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13569

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₂×C₇×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#124 ◁ G

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

normal subgroups only

This group has 492 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 6557);