C₇×#11811 #13189027

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11811

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁴×C₇ ◁ C₂²×C₇×C₂²⋊C₄ ◁ G

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

normal subgroups only

This group has 496 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)(10,18,21,35)(11,23,26,16)(12,15)(13,17)(14,20,29,24)(19,36,22,33)(25,32)(27,34,39,31)(28,38,30,37);(8,10)(9,14,17,30)(11,24)(12,22)(13,19)(15,29,32,28)(16,33,23,35)(18,34,36,31)(20,27)(21,25)(26,38)(37,39);(9,15)(10,19)(14,28)(16,31)(17,32)(18,35)(20,37)(21,22)(23,34)(24,38)(29,30)(33,36)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4799);