C₇×#2101 #11730570

order	896
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2101

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 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)(8,9,12,14)(10,15,11,13)(16,17)(18,20)(19,21)(22,23)(24,25,27,28)(26,29,31,30);(8,10,12,11)(9,13,14,15)(16,18,19,22)(17,20,21,23)(24,26,27,31)(25,29,28,30);(8,11,12,10)(9,13,14,15)(17,21)(20,23)(25,28)(26,30)(29,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2101, #13383714, #13383723, …
maximal subgroup of	C₂×C₇×#2101, C₇×#23744, C₇×#23746, …
maximal normal subgroup of	C₂×C₇×#2101, C₇×#23744, C₇×#23746, …
maximal quotient of	C₇×#10689, C₇×#11208, C₇×#11268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5336);