C₇×#19618 #13196834

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#19618

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×#127
prime radical	C₂³×C₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

690 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 278 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) = 896

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

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

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, 12606);