C₇×#12167 #13189383

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12167

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₇×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#424 ◁ G

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 896

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

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

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, 5155);