C₇×#9460 #13186676

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#9460

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

512 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 448

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2448);