C₇×#1468 #11729937

order	896
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#1468

Subgroups

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

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 224

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13246121, #13246128, #13246390, …
maximal subgroup of	C₇×#8487, C₇×#8488, C₇×#8539, …
maximal normal subgroup of	C₇×#8487, C₇×#8488, C₇×#8539, …
maximal quotient of	C₇×#7654, C₇×#7655, C₇×#7658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4703);