C₇×#2235 #11730704

order	896
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#2235

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₂³×C₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

140 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 64 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)(16,17,20,24)(18,21,27,30)(19,28,26,23)(22,31,29,25);(9,12)(13,15);(8,10,11,14)(9,13,12,15)(17,21)(18,25)(19,26)(22,23)(24,30)(27,31)(28,29)

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13267000, #13267007, #13267038, …
maximal subgroup of	C₇×#12279, C₇×#12280, C₇×#12285, …
maximal normal subgroup of	C₇×#12279, C₇×#12280, C₇×#12285, …
maximal quotient of	C₇×#12866, C₇×#12920, C₇×#12993, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5470);