C₇×#18731 #13195947

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#18731

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

884 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 11719);