C₇×#12468 #13189684

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12468

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#1667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×D₄ ◁ G

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

normal subgroups only

This group has 234 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)(8,9)(10,12)(11,13)(14,15)(17,21)(18,25)(19,26)(22,27)(23,31)(24,30)(28,29);(9,12)(13,14)(16,17,22,31,19,23,18,21,20,24,25,29,26,28,27,30);(9,13)(11,15)(12,14)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5456);