C₇×#10317 #13187533

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#10317

Subgroups

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

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

normal subgroups only

This group has 554 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) = 1792

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

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

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, 3305);