C₇×#10522 #13187738

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10522

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

402 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9)(10,13)(11,14)(12,15)(16,17,21,25)(18,26,23,19)(20,27,30,24)(22,29,31,28);(8,9)(10,14)(11,13)(12,15)(16,18,21,23)(17,22,25,31)(19,27,26,24)(20,29,30,28);(9,13)(10,11)(14,15)(16,19,20,22,21,26,30,31)(17,23,24,29,25,18,27,28)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3510);