C₇×#19006 #13196222

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#19006

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#562	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

938 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 11994);