C₇×#11391 #13188607

order	1792
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#11391

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#120
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#379
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ G

842 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 254 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)(16,17,20,25,21,24,33,38)(18,26,29,39,28,22,32,35)(19,23,31,36,30,37,43,47)(27,34,42,45,41,40,44,46);(8,9,11,13)(10,14,15,12)(16,18)(17,22)(19,27)(20,32)(21,28)(23,34)(24,26)(25,39)(29,33)(30,41)(31,44)(35,38)(36,46)(37,40)(42,43)(45,47);(8,10,11,15)(9,12,13,14)(16,19)(17,23)(18,27)(20,31)(21,30)(22,34)(24,37)(25,36)(26,40)(28,41)(29,42)(32,44)(33,43)(35,45)(38,47)(39,46)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4379);