C₇×#9016 #13186232

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#9016

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

836 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 35
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2004);