C₇×#12809 #13190025

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12809

Subgroups

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1799
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₂×C₇×#131 ◁ G

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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)(8,9,13,17,14,19,11,25)(10,20,23,31,24,36,21,15)(12,18,26,32,27,16,28,33)(22,35,37,30,38,34,39,29);(8,10,11,21,14,24,13,23)(9,15,17,20,19,31,25,36)(12,22,26,37,27,38,28,39)(16,29,32,34,18,30,33,35);(9,16)(12,27)(15,29)(17,32)(18,19)(20,34)(22,38)(25,33)(26,28)(30,31)(35,36)(37,39)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5797);