C₇×#13598 #13190814

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13598

Subgroups

	subgroup	G/H
center	C₄×C₇	#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	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

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

normal subgroups only

This group has 318 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)(8,9,12,17,13,18,28,36)(10,19,22,37,21,30,38,35)(11,15,24,33,25,16,27,34)(14,29,31,23,32,26,39,20);(8,10,13,21)(9,14)(11,20,25,23)(12,26,28,29)(15,30)(16,19)(17,35)(18,32)(22,27,38,24)(31,34)(33,39)(36,37);(8,11,13,25)(9,15,18,16)(10,20,21,23)(12,24,28,27)(14,30,32,19)(17,33,36,34)(22,29,38,26)(31,35,39,37)

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, 6586);