C₇×#12067 #13189283

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12067

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#442
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#322 ◁ G

246 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,9,11,14,12,10,13,15)(16,17,19,22,20,25,30,24)(18,26,29,23,28,21,27,31);(17,21)(18,27)(19,30)(22,23)(24,31)(25,26)(28,29);(8,10,11,15,12,9,13,14)(17,22,25,24)(18,28)(21,23,26,31)(27,29)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#12067
maximal subgroup of	D₇×#12067
maximal normal subgroup of	D₇×#12067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5055);