C₇×#13599 #13190815

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13599

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₂³×C₇	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₄⋊Q₈ ◁ G

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

normal subgroups only

This group has 312 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)(8,9,11,16,13,18,25,35)(10,19,21,37,20,36,38,33)(12,17,27,34,28,15,24,32)(14,29,31,22,30,23,39,26);(8,10,13,20)(9,14)(11,23,25,29)(12,26,28,22)(15,19)(16,33)(17,36)(18,30)(21,24,38,27)(31,32)(34,39)(35,37);(10,20)(14,30)(19,36)(21,38)(22,26)(23,29)(31,39)(33,37)

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, 6587);