C₇×#13342 #13190558

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13342

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₇×#3150	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

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

normal subgroups only

This group has 558 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)(12,13,20,27)(14,22,32,42)(15,33,28,21)(16,24,36,43)(17,23,34,37)(18,26,19,25)(29,40,35,39)(30,41,31,38);(8,9)(10,11)(12,14)(13,21)(15,34)(16,35)(17,28)(18,31)(19,30)(20,32)(22,24)(23,41)(25,40)(26,39)(27,33)(29,36)(37,38)(42,43);(9,11)(12,15)(13,22)(14,16)(17,30)(18,29)(19,35)(20,28)(21,37)(23,33)(24,40)(25,41)(26,38)(27,42)(31,34)(32,36)(39,43)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6330);