C₇×#13596 #13190812

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13596

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6584);