C₇×#10080 #13187296

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10080

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 552 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) = 35
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 35 points:

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

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, 3068);