C₇×#1700 #11730169

order	896
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#1700

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

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

normal subgroups only

This group has 146 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) = 896

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

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

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1700, #13336478, #13336497, …
maximal subgroup of	C₂×C₇×#1700, C₇×#19155, C₇×#19156, …
maximal normal subgroup of	C₂×C₇×#1700, C₇×#19155, C₇×#19156, …
maximal quotient of	C₇×#8655, C₇×#9226, C₇×#9228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4935);