C₇×#13333 #13190549

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13333

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#459
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

546 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 122 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)(8,9)(10,11)(12,13)(14,25)(15,31)(16,34)(17,23)(18,37)(19,24)(20,27)(21,30)(22,26)(28,40)(29,39)(32,38)(33,42)(35,41)(36,43);(9,11)(12,14,19,30)(13,20,24,38)(15,16,33,36)(17,29,18,28)(21,34,25,43)(22,40,41,39)(23,42,37,31)(26,27,35,32);(12,15,19,33)(13,21,24,25)(14,26,30,35)(16,28,36,29)(17,27,18,32)(20,22,38,41)(23,40,37,39)(31,43,42,34)

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, 6321);