C₇×#12391 #13189607

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12391

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#446 ◁ G

290 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5379);