C₇×#9390 #13186606

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9390

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

446 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 448

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

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

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, 2378);