C₇×#3249 #11742558

order	896
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#3249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

348 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(9,12)(10,14)(16,17,20,23)(18,24,26,31)(19,27,29,22)(21,28,30,25);(8,9)(10,13)(11,12)(14,15);(9,12)(13,15)(16,18,20,26)(17,21,23,30)(19,28,29,25)(22,31,27,24);(8,10,11,14)(9,13,12,15)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13254750, #13254755, #13255887, …
maximal subgroup of	C₇×#10030, C₇×#10031, C₇×#10195, …
maximal normal subgroup of	C₇×#10030, C₇×#10031, C₇×#10195, …
maximal quotient of	C₇×#21885, C₇×#21961, C₇×#22113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 17324);