C₇×#13609 #13190825

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13609

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#3220	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#535 ◁ G

348 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 86 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)(10,19,21,14)(11,15)(12,17,27,16)(13,18)(20,34,35,28)(22,29,26,30)(23,36,37,39)(24,32)(25,31,38,33);(8,10)(9,14)(11,23)(12,26)(13,22)(15,28)(16,29)(17,19)(18,30)(20,24)(21,27)(25,35)(31,34)(32,39)(33,36)(37,38);(8,11)(9,15)(10,20)(12,25)(13,24)(14,28)(16,33)(17,31)(18,32)(19,34)(21,35)(22,23)(26,37)(27,38)(29,36)(30,39)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6597);