C₇×#10225 #13187441

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#10225

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

574 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 122 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)(10,13)(11,14)(12,15)(16,17,22,27)(18,28,31,39)(19,24,35,42)(20,26,21,25)(23,32,38,30)(29,44,45,40)(33,47,46,37)(34,43,36,41);(9,13)(10,11)(14,15)(16,18)(17,23)(19,33)(20,32)(21,30)(22,31)(24,40)(25,39)(26,28)(27,38)(29,34)(35,46)(36,45)(37,41)(42,44)(43,47);(9,14)(10,11)(13,15)(16,19)(17,24)(18,29)(20,34)(21,36)(22,35)(23,37)(25,41)(26,43)(27,42)(28,40)(30,33)(31,45)(32,46)(38,47)(39,44)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3213);