C₇×#2163 #11730632

order	896
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2163

Subgroups

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

152 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 104 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)(16,17,18,22,19,23,24,29,20,21,25,27,26,28,30,31);(9,12)(13,15);(8,10,11,14)(9,13,12,15)(17,21)(22,27)(23,28)(29,31)

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13240639, #13240643, #13240646, …
maximal subgroup of	C₇×#7143, C₇×#7144, C₇×#7146, …
maximal normal subgroup of	C₇×#7143, C₇×#7144, C₇×#7146, …
maximal quotient of	C₇×#12591, C₇×#12599, C₇×#12653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5398);