C₇×#13570 #13190786

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

Aliases

direct product

C₇×#13570

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	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

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

normal subgroups only

This group has 482 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)(10,20,23,30)(11,16)(12,27)(13,18,29,35)(14,19)(15,22,31,39)(17,28,37,26)(21,32,38,33)(24,34,25,36);(8,10)(9,15)(11,22)(12,21)(13,23)(14,25)(16,34)(17,36)(18,31)(19,33)(20,37)(24,29)(26,39)(27,30)(28,38)(32,35);(8,11,14,12)(9,16,19,27)(10,21,25,22)(13,26,29,28)(15,30,33,34)(17,35,37,18)(20,32,36,31)(23,38,24,39)

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, 6558);