C₇×#8491 #13185707

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#8491

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

820 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 1479);