C₇×#1626 #11730095

order	896
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1626

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

210 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(17,21)(18,22)(19,23);(8,9)(10,13)(11,15)(12,14)(16,17,20,18)(19,22,23,21);(8,10,12,11)(9,13,14,15)(16,18,20,17)(19,21,23,22)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1626, #13329555, #13329775, …
maximal subgroup of	C₂×C₇×#1626, C₇×#18494, C₇×#18518, …
maximal normal subgroup of	C₂×C₇×#1626, C₇×#18494, C₇×#18518, …
maximal quotient of	C₇×#7739, C₇×#7743, C₇×#7747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4861);