C₇×#32876 #13210092

order	1792
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#32876

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

584 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 186 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)(8,9,13,17)(10,18,20,14)(11,22,24,15)(12,26,28,16)(19,34,35,29)(21,30,27,33)(23,31,38,36)(25,32,39,37);(8,10,12,27,13,20,28,21)(9,14,16,33,17,18,26,30)(11,23,25,19,24,38,39,35)(15,31,32,29,22,36,37,34);(8,11,12,25,13,24,28,39)(9,15,16,32,17,22,26,37)(10,19,21,23,20,35,27,38)(14,29,30,31,18,34,33,36);(19,35)(23,38)(29,34)(31,36)

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, 25864);