C₇×#13566 #13190782

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13566

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

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

normal subgroups only

This group has 478 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)(8,9,12,18,14,19,27,33)(10,20,23,31,25,37,39,15)(11,16,26,17,13,28,29,34)(21,35,38,32,24,36,22,30);(8,10)(9,15)(11,24)(12,22)(13,21)(14,25)(16,32)(17,20)(18,35)(19,31)(23,29)(26,39)(27,38)(28,30)(33,36)(34,37);(8,11,14,13)(9,16,19,28)(10,21,25,24)(12,26,27,29)(15,30,31,32)(17,33,34,18)(20,35,37,36)(22,23,38,39)

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, 6554);