C₇×#21827 #13199043

order	1792
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#21827

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×C₂²⋊C₄ ◁ G

1130 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 366 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)(10,18)(11,22,25,15)(12,16)(13,17)(14,21)(19,33,34,30)(20,28)(23,37,38,27)(24,36,39,31)(26,32)(29,35);(8,10,12,21)(9,14,16,18)(11,23,24,19)(13,20,26,35)(15,30,31,27)(17,29,32,28)(22,33,36,37)(25,38,39,34);(8,11,12,24)(9,15,16,31)(10,19,21,23)(13,25,26,39)(14,27,18,30)(17,22,32,36)(20,34,35,38)(28,33,29,37);(9,16)(10,20)(14,28)(15,31)(17,32)(18,29)(19,34)(21,35)(22,36)(23,38)(27,33)(30,37)

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, 14815);