C₇×#7441 #13184657

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#7441

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1353
commutator	C₂×M₁₆	C₂×C₄×C₇
Frattini, cosocle	#423	C₂²×C₇
prime radical	C₂³×C₇	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#423 ◁ G

136 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 39 points:

(1,2,3,4,5,6,7)(8,9)(10,18,20,30,22,35,21,38)(11,15,24,31)(12,26)(13,16)(14,19,28,36,29,37,34,23)(17,27)(25,33,39,32);(8,10)(9,14,16,29)(11,23,24,36)(12,21)(13,22)(15,30)(17,28,26,34)(18,33)(19,25,37,39)(20,27)(31,38)(32,35)

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, 429);