C₇×#1805 #11730274

order	896
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#1805

Subgroups

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

292 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13245771, #13245785, #13245799, …
maximal subgroup of	C₇×#8429, C₇×#8431, C₇×#8433, …
maximal normal subgroup of	C₇×#8429, C₇×#8431, C₇×#8433, …
maximal quotient of	C₇×#8566, C₇×#8570, C₇×#8574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5040);