C₇×#12230 #13189446

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#12230

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₇×#392	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₂²⋊C₄ ◁ G

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

normal subgroups only

This group has 422 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,12,17)(10,15,21,30)(11,14,22,16)(13,18,25,33)(19,31,34,39)(20,26,37,29)(23,28,35,38)(24,27,36,32);(9,14)(10,19)(11,22)(15,28)(16,17)(18,27)(20,35)(21,34)(23,37)(24,36)(26,31)(29,39)(30,38)(32,33);(8,10,12,21)(9,15,17,30)(11,23,22,35)(13,19,25,34)(14,26,16,29)(18,31,33,39)(20,36,37,24)(27,28,32,38)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5218);