C₇×#10025 #13187241

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10025

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

470 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 98 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)(8,9,13,11)(10,16,19,23)(12,20,22,15)(14,21,17,18)(24,25)(26,32)(27,35)(28,31)(29,34)(30,37)(33,39)(36,38);(8,10,13,19)(9,14,11,17)(12,21,22,18)(15,23,20,16)(29,38)(32,39);(8,11,13,9)(10,17,19,14)(12,15,22,20)(16,21,23,18)(24,26)(25,29)(27,36)(28,33)(30,39)(31,38)(32,35)(34,37)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3013);