C₇×#12376 #13189592

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12376

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3273	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#436 ◁ G

366 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,12,15)(10,13,11,14)(16,17,21,25)(18,26,28,44)(19,30,32,46)(20,35,33,24)(22,37,38,27)(23,40,42,34)(29,45,36,41)(31,43,47,39);(9,13)(10,11)(14,15)(16,18)(17,22)(19,31)(20,36)(21,28)(23,41)(24,43)(25,38)(26,30)(27,34)(29,33)(32,47)(35,39)(37,40)(42,45)(44,46);(9,14)(10,11)(13,15)(19,32)(23,42)(26,44)(27,37)(30,46)(31,47)(34,40)(41,45)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5364);