C₇×#13611 #13190827

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13611

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₇×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#536 ◁ G

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

normal subgroups only

This group has 316 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)(8,9,11,17,13,18,24,32)(10,15,20,28,19,29,33,39)(12,14,22,27,25,16,23,31)(21,26,36,37,34,30,35,38);(9,14)(10,19)(11,22)(12,25)(15,26)(16,18)(20,35)(23,24)(27,31)(29,30)(33,36)(37,38);(8,10)(9,15)(11,20)(12,21)(13,19)(14,26)(16,30)(17,28)(18,29)(22,36)(23,35)(24,33)(25,34)(27,37)(31,38)(32,39)

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, 6599);