C₇×#13372 #13190588

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13372

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 370 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) = 43
transitive degree	μ_t(G) = 448

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

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

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, 6360);