C₇×#12360 #13189576

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12360

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
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×#446 ◁ G

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

normal subgroups only

This group has 308 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)(9,13)(10,14)(11,15)(17,22)(18,21)(19,26)(23,30)(24,31)(25,28)(27,29);(8,9,12,10)(11,14,15,13)(16,17,20,24)(18,23,21,29)(19,27,28,30)(22,25,31,26);(8,10,12,9)(11,13,15,14)(16,18,20,21)(17,23,24,29)(19,26,28,25)(22,30,31,27)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5348);