C₇×#11426 #13188642

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#11426

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂³×C₇×D₄	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁴×C₇ ◁ C₂²×C₇×C₂²⋊C₄ ◁ G

734 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 31 points:

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

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, 4414);