C₇×#2863 #11742172

order	896
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#2863

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

408 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13242195, #13242215, #13242287, …
maximal subgroup of	C₇×#7669, C₇×#7673, C₇×#7689, …
maximal normal subgroup of	C₇×#7669, C₇×#7673, C₇×#7689, …
maximal quotient of	C₇×#14688, C₇×#14699, C₇×#14703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16938);