C₇×#14678 #13191894

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₇×#14678

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

1182 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 462 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) = 1792

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(17,20)(21,23)(24,25,28,26)(27,29,31,30);(9,11)(12,13,14,15)(17,20)(18,22)(25,26)(27,31);(8,9,10,11)(16,17,19,20)(18,21,22,23)(24,26,28,25)(27,29,31,30);(16,18,19,22)(17,21,20,23)(24,27,28,31)(25,29,26,30)

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, 7666);