C₇×#1649 #11730118

order	896
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#1649

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₂×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 31 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1649, #13329644, #13329753, …
maximal subgroup of	C₂×C₇×#1649, C₇×#18503, C₇×#18516, …
maximal normal subgroup of	C₂×C₇×#1649, C₇×#18503, C₇×#18516, …
maximal quotient of	C₇×#7756, C₇×#7757, C₇×#7763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4884);