C₇×#13680 #13190896

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

Aliases

direct product

C₇×#13680

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#3289	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#124 ◁ G

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

normal subgroups only

This group has 378 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)(8,9,15,20)(10,21,24,25)(11,22,16,28)(12,27,30,17)(13,18,32,37)(14,33,35,19)(23,39,38,29)(26,36,34,31);(8,10)(9,16)(11,20)(12,28)(13,31)(14,23)(15,24)(17,25)(18,29)(19,34)(21,27)(22,30)(26,33)(32,36)(35,38)(37,39);(8,11,15,16)(9,10,20,24)(12,29,30,39)(13,25,32,21)(14,34,35,26)(17,36,27,31)(18,22,37,28)(19,38,33,23)

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, 6668);