C₇×#1495 #11729964

order	896
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1495

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₄²	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₇×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,12)(10,14)(13,15)(16,17,19,23,20,22,26,29)(18,21,25,27,24,28,30,31);(8,9)(10,15)(11,13)(12,14)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(9,13)(12,15)(17,22)(18,24)(23,29)(25,30)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13246096, #13246103, #13246110, …
maximal subgroup of	C₇×#8483, C₇×#8484, C₇×#8485, …
maximal normal subgroup of	C₇×#8483, C₇×#8484, C₇×#8485, …
maximal quotient of	C₇×#7694, C₇×#7695, C₇×#7696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4730);