C₇×#13292 #13190508

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13292

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂²×D₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

484 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6280);