C₇×#2980 #11742289

order	896
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2980

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂³×C₄
prime radical	C₇×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

394 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 248 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 — 4 generators on 31 points:

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

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13248199, #13248745, #13248755, …
maximal subgroup of	C₇×#8918, C₇×#9005, C₇×#9007, …
maximal normal subgroup of	C₇×#8918, C₇×#9005, C₇×#9007, …
maximal quotient of	C₇×#20532, C₇×#20539, C₇×#20563, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 17055);