C₇×#2028 #11730497

order	896
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#2028

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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)(8,9,11,13)(10,14,15,12)(16,17)(18,24)(19,27)(20,23)(21,25)(22,28)(26,31)(29,30);(8,10,11,15)(9,12,13,14)(19,28)(21,30)(22,27)(25,29);(16,18,20,26)(17,21,23,30)(19,29,28,25)(22,31,27,24)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241390, #13266556, #13266570, …
maximal subgroup of	C₇×#7431, C₇×#12210, C₇×#12212, …
maximal normal subgroup of	C₇×#7431, C₇×#12210, C₇×#12212, …
maximal quotient of	C₇×#10341, C₇×#10342, C₇×#10345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5263);