C₇×#11896 #13189112

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11896

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

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

normal subgroups only

This group has 390 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) = 448

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

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

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, 4884);