C₇×#7076 #13184292

order	1792
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7076

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#327
commutator	C₂²×C₈	C₂×C₄×C₇
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₇
prime radical	C₇×#1999	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

338 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 31 points:

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

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, 64);