C₇×#13594 #13190810

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13594

Subgroups

	subgroup	G/H
center	C₄×C₇	#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
socle	C₂×C₇	C₂×#354
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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)(10,19,22,29)(11,15)(12,17,26,16)(13,18)(14,21,28,25)(20,33,35,39)(23,31,37,30)(24,32)(27,34,38,36);(8,10,13,21)(9,14)(11,20,24,34)(12,25,26,22)(15,27)(16,19)(17,28)(18,29)(23,36,37,35)(30,33)(31,38)(32,39);(8,11,13,24)(9,15,18,32)(10,20,21,34)(12,23,26,37)(14,27,29,39)(16,30,17,31)(19,33,28,38)(22,35,25,36)

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, 6582);