C₇×#12428 #13189644

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12428

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#2206	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈² ◁ G

256 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,12)(10,13)(14,15)(16,17)(18,21)(19,22)(20,28)(23,27)(24,26)(25,29)(30,31);(8,9,13,15,11,14,10,12)(17,21)(22,29)(23,30)(24,28);(17,22)(19,27)(20,26)(21,29)(23,24)(25,31)(28,30)

Conjugacy classes & character table

301 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5416);