C₇×#11851 #13189067

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11851

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₇×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

418 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,11,17,12,18,24,35)(10,16,21,33,20,28,23,34)(13,15,25,31,26,32,22,37)(14,27,29,36,30,38,39,19);(9,14)(10,19)(11,22)(15,16)(17,33)(18,30)(20,36)(21,23)(24,25)(27,38)(28,32)(29,31)(34,35)(37,39);(9,15)(10,20)(14,28)(16,30)(17,31)(18,32)(19,36)(21,23)(27,38)(29,34)(33,39)(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, 4839);