C₇×#13327 #13190543

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13327

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

594 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 146 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)(10,11)(17,22)(18,24)(19,26)(20,29)(23,28)(25,27)(30,31);(9,11)(12,13)(14,15)(16,17,21,18)(19,27,30,28)(20,25,29,23)(22,31,24,26);(13,15)(16,18,21,17)(19,28,30,27)(20,23,29,25)(22,26,24,31)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6315);