C₇×#12030 #13189246

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12030

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

218 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5018);