C₇×#12865 #13190081

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12865

Subgroups

	subgroup	G/H
center	C₄×C₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1820
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×#41 ◁ G

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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)(10,19,22,17)(11,23)(12,16)(13,18)(14,21,29,36)(15,20,32,24)(25,28)(26,34)(27,33)(30,39)(31,35,37,38);(8,10)(9,14,18,29)(11,24)(12,21)(13,22)(15,30,32,28)(16,19,33,17)(20,26)(23,37,34,31)(25,38)(27,36)(35,39);(8,11,12,25,13,26,27,39)(9,15,16,31,18,32,33,37)(10,20,21,35,22,24,36,38)(14,28,19,34,29,30,17,23)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5853);