C₇×#1611 #11730080

order	896
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#1611

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

210 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 224

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

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13266550, #13266557, #13266564, …
maximal subgroup of	C₇×#12209, C₇×#12210, C₇×#12211, …
maximal normal subgroup of	C₇×#12209, C₇×#12210, C₇×#12211, …
maximal quotient of	C₇×#7794, C₇×#7798, C₇×#7807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4846);