C₇×#13562 #13190778

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13562

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

490 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6550);