C₇×#2228 #11730697

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

Aliases

direct product

C₇×#2228

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₂×C₇×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 136 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)(8,9,11,13)(10,14,15,12)(16,17,18,21)(19,22,25,29)(20,28,26,24)(23,31,30,27);(8,10,11,15)(9,12,13,14)(17,21)(22,29)(23,30)(24,28);(17,22)(19,27)(20,26)(21,29)(23,24)(25,31)(28,30)

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2228, #13508439, #13508449, …
maximal subgroup of	C₂×C₇×#2228, C₇×#33626, C₇×#33627, …
maximal normal subgroup of	C₂×C₇×#2228, C₇×#33626, C₇×#33627, …
maximal quotient of	C₇×#12813, C₇×#12935, C₇×#12937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5463);