C₇×#23795 #13201011

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#23795

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	#2397
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

908 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 4 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,12,16,13,17,25,31)(10,18,20,26,21,30,24,14)(11,15,23,29,22,28,35,38)(19,32,34,37,33,39,36,27);(8,10)(9,14)(11,19)(12,24)(13,21)(15,27)(16,30)(17,26)(18,31)(20,25)(22,33)(23,36)(28,37)(29,39)(32,38)(34,35);(11,22)(14,26)(15,28)(18,30)(19,33)(23,35)(29,38)(34,36);(8,11,12,23,13,22,25,35)(9,15,16,29,17,28,31,38)(10,19,24,36,21,33,20,34)(14,27,30,39,26,37,18,32)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 16783);