C₇×#12280 #13189496

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12280

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₇×#3191	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₂²⋊C₈ ◁ G

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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)(10,13,17,20)(11,14,18,21)(16,22,23,19)(25,30)(26,29)(27,34)(31,39)(32,38)(33,36)(35,37);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,23)(21,22)(24,25)(26,31)(27,35)(28,32)(29,37)(30,34)(33,38)(36,39);(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,23)(20,22)(24,26,28,29)(25,31,32,37)(27,33,36,34)(30,35,38,39)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5268);