C₇×#2858 #11742167

order	896
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₇×#2858

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

752 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 264 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 — 4 generators on 23 points:

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13242152, #13242244, #13242336, …
maximal subgroup of	C₇×#7659, C₇×#7679, C₇×#7699, …
maximal normal subgroup of	C₇×#7659, C₇×#7679, C₇×#7699, …
maximal quotient of	C₇×#14687, C₇×#14700, C₇×#14708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16933);