C₇×#7146 #13184362

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7146

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#95
commutator	C₂×C₄	C₄×C₇×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇
socle	C₂²×C₇	#336
prime radical	C₂³×C₇	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 896

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 134);