C₇×#1633 #11730102

order	896
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1633

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

206 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241633, #13274569, D₇×#1633, …
maximal subgroup of	C₇×#7522, C₇×#13684, C₂×C₇×#1633, …
maximal normal subgroup of	C₇×#7522, C₇×#13684, C₂×C₇×#1633, …
maximal quotient of	C₇×#7757, C₇×#7759, C₇×#7784, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4868);