C₇×#13347 #13190563

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13347

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₇
prime radical	C₂×C₇×#453	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₇×D₄ ◁ G

830 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6335);