C₇×#10319 #13187535

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

Aliases

direct product

C₇×#10319

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

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 43
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(13,18)(15,16)(19,22)(20,21)(23,24)(26,27)(28,29,33,37)(30,38,41,42)(31,36,32,35)(34,40,43,39);(9,11)(12,13,17,21)(14,22,25,26)(15,20,16,18)(19,23,27,24)(28,30,33,41)(29,34,37,43)(31,40,32,39)(35,42,36,38);(12,14,17,25)(13,19,21,27)(15,24,16,23)(18,22,20,26)(29,35)(31,32)(34,42)(36,37)(38,43)(39,40)

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, 3307);