C₇×#10306 #13187522

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#10306

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#127
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

670 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 170 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)(8,9)(10,11)(12,13)(14,16)(15,18)(17,19)(20,21,25,29)(22,30,33,34)(23,28,24,27)(26,32,35,31);(9,11)(12,13)(14,16)(15,18)(17,19)(20,22,25,33)(21,26,29,35)(23,32,24,31)(27,34,28,30);(13,16)(14,19)(17,18)(22,31)(23,24)(26,30)(27,28)(32,33)(34,35)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3294);