C₇×#2822 #11742131

order	896
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2822

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

380 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 35
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3,4,5,6,7)(13,14)(17,18)(20,21,25,27)(22,28,30,23)(24,26,34,35)(29,33,32,31);(9,11)(12,13,16,14)(15,17,19,18)(20,22,25,30)(21,23,27,28)(24,32,34,29)(26,33,35,31);(8,9)(10,11)(12,14,16,13)(15,17,19,18)(20,23)(21,22)(24,33)(25,28)(26,32)(27,30)(29,35)(31,34);(12,15,16,19)(13,17,14,18)(20,24,25,34)(21,26,27,35)(22,29,30,32)(23,31,28,33)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13255087, #13255114, #13255132, …
maximal subgroup of	C₇×#10080, C₇×#10084, C₇×#10087, …
maximal normal subgroup of	C₇×#10080, C₇×#10084, C₇×#10087, …
maximal quotient of	C₇×#14629, C₇×#14631, C₇×#14634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16897);