C₇×#13679 #13190895

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13679

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#3288	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#124 ◁ G

386 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 58 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)(8,9,13,19)(10,20,24,37)(11,17,27,16)(12,18,30,31)(14,25,34,21)(15,35,33,22)(23,32,28,36)(26,39,29,38);(8,10)(9,14)(11,25)(12,23)(13,24)(15,31)(16,37)(17,20)(18,33)(19,34)(21,27)(22,29)(26,35)(28,30)(32,38)(36,39);(8,10)(9,15)(11,22)(12,28)(13,24)(14,31)(16,37)(17,20)(18,34)(19,33)(21,26)(23,30)(25,29)(27,35)(32,39)(36,38)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6667);