C₇×#23840 #13201056

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#23840

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

662 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 1792

A minimal faithful permutation representation — 4 generators on 39 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 16828);