C₇×#2232 #11730701

order	896
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2232

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₇×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

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

normal subgroups only

This group has 150 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)(10,13)(11,12)(14,15)(16,17)(18,21)(19,28)(20,24)(22,25)(23,26)(27,30)(29,31);(9,12)(13,15);(8,10,11,14)(9,13,12,15)(17,21)(18,25)(19,26)(22,23)(24,30)(27,31)(28,29)

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13266995, #13267005, #13267031, …
maximal subgroup of	C₇×#12278, C₇×#12280, C₇×#12284, …
maximal normal subgroup of	C₇×#12278, C₇×#12280, C₇×#12284, …
maximal quotient of	C₇×#12866, C₇×#12919, C₇×#12991, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5467);