C₇×#11960 #13189176

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11960

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

392 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,10,13,11,12,14,15)(16,17,21,27,22,25,37,43,23,28,35,45,19,34,20,26)(18,29,32,42,30,44,31,39,33,46,36,24,38,40,47,41);(9,12)(10,14)(16,18)(17,24)(19,30)(20,36)(21,31)(22,38)(23,33)(25,39)(26,44)(27,29)(28,42)(32,37)(34,41)(35,47)(40,43)(45,46);(9,12)(13,15)(16,19,23,22)(17,25,28,34)(18,30,33,38)(20,35,37,21)(24,39,42,41)(26,27,43,45)(29,40,46,44)(31,36,47,32)

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, 4948);