C₇×#12146 #13189362

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12146

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₇×#537	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₄⋊C₄ ◁ G

380 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 70 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)(9,13)(10,11)(14,15)(16,17)(18,21)(19,22)(20,24)(23,28)(25,30)(26,31)(27,29);(8,9)(10,13)(11,15)(12,14)(16,18,20,25)(17,21,24,30)(19,27,28,26)(22,31,23,29);(8,10,12,11)(9,13,14,15)(17,22)(18,25)(19,28)(21,29)(23,24)(30,31)

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, 5134);