C₇×#15155 #13192371

order	1792
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₇×#15155

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

2158 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 2158 conjugacy classes of subgroups — too many to draw. Only its 558 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) = 27
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 8143);