C₇×#11965 #13189181

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×#11965

Subgroups

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

382 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 126 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)(16,17,20,25,22,27,35,24,21,26,36,43,34,42,19,33)(18,28,30,39,32,41,46,45,31,44,47,23,37,38,29,40);(9,12)(10,14)(16,18,21,31)(17,23,26,39)(19,30,35,47)(20,29,36,46)(22,37,34,32)(24,41,33,38)(25,44,43,28)(27,45,42,40);(9,12)(13,15)

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, 4953);