C₇×#2078 #11730547

order	896
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2078

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

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

normal subgroups only

This group has 198 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)(10,13)(11,12)(14,15)(16,17,19,24,21,25,20,23)(18,22,27,30,28,29,26,31);(9,12)(13,15)(16,18)(17,22)(19,27)(20,26)(21,28)(23,31)(24,30)(25,29);(8,10,11,14)(9,13,12,15)(17,23)(18,26)(19,20)(22,29)(24,25)(27,28)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13267030, #13267037, #13267044, …
maximal subgroup of	C₇×#12284, C₇×#12285, C₇×#12286, …
maximal normal subgroup of	C₇×#12284, C₇×#12285, C₇×#12286, …
maximal quotient of	C₇×#10615, C₇×#10616, C₇×#10617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5313);