C₇×#12062 #13189278

order	1792
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12062

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₈×D₄
prime radical	C₂×C₇×D₈	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5050);