C₇×#12874 #13190090

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12874

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1823
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

376 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 5862);